Lesson OVERVIEW of lessons on Miscellaneous adavanced Geometry problems

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<H2>OVERVIEW of lessons on Miscellaneous advanced Geometry problems</H2>

My lessons on Miscellaneous advanced Geometry problems in this site are

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&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=https://www.algebra.com/algebra/homework/word/geometry/Triangle-with-the-side-ratio-456-has-a-smallest-angle-half-of-the-biggest-angle.lesson>Triangle with the sides ratio 4:5:6 has the smallest angle measure half of the biggest angle</A> &nbsp;&nbsp;

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=https://www.algebra.com/algebra/homework/word/geometry/Advanced-problem-on-equilateral-triangles-built-externally-on-sides-of-a-triangle-.lesson>Advanced problem on equilateral triangles built externally on sides of an arbitrary triangle</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=https://www.algebra.com/algebra/homework/word/geometry/Advanced-problem-on-squares-built-externally-on-sides-of-a-triangle.lesson>Advanced problem on squares built externally on sides of an arbitrary triangle</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=https://www.algebra.com/algebra/homework/word/geometry/Selected-problems-from-the-archive-on-the-area-of-plane-shapes.lesson>Selected problems from the archive on the area of plane shapes</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=https://www.algebra.com/algebra/homework/word/geometry/Two-unit-squares-sharing-the-same-center-but-turned-%28rotated%29-each-relative-other.lesson>Two unit squares sharing the same center but turned (rotated) each relative the other</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=https://www.algebra.com/algebra/homework/word/geometry/Finding-the-hypotenuse-of-a-right-angled-triangle-via-its-two-medians.lesson>Finding the hypotenuse of a right-angled triangle via its two medians</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=https://www.algebra.com/algebra/homework/word/geometry/Area-of-a-triangle-obtained-by-cutting-uniform-strips-from-a-given-triangle.lesson>Area of a triangle obtained by cutting uniform strips from the given triangle</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=https://www.algebra.com/algebra/homework/word/geometry/Perimeter-of-a-thriangle-obtained-by-adding-uniform-strip-to-a-given-triangle.lesson>Find the perimeter of a triangle obtained by adding uniform strip to a given triangle</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=https://www.algebra.com/algebra/homework/word/geometry/Center-of-the-given-circle-is-the-incenter-of-the-given-triangle.lesson>Center of the given circle is the incenter of the given triangle</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=https://www.algebra.com/algebra/homework/word/geometry/Determine-the-standard-form-eqn-of-the-circle-inscribed-in-a-triangle.lesson>Determine the standard form equation of the circle inscribed in a triangle</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=https://www.algebra.com/algebra/homework/word/geometry/Find-the-side-of-a-square-if-distances-are-given-from-an-interior-point-to-3-its-vertices.lesson>Find the side of a square if distances are given from an interior point to 3 its vertices</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=https://www.algebra.com/algebra/homework/word/geometry/The-point-which-minimizes-the-sum-of-distances-to-vertices-of-a-given-quadrilateral.lesson>The point which minimizes the sum of distances to vertices of a given quadrilateral</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=https://www.algebra.com/algebra/homework/word/geometry/Finding-surface-area-of-the-rectangular-box.lesson>Problems on surface area of a rectangular box</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=https://www.algebra.com/algebra/homework/word/geometry/Find-the-volume-of-a-rectangular-box-if-the-areas-of-its-faces-are-given.lesson>Find the volume and the dimensions of a rectangular box if the areas of its faces are given</A>

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&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=https://www.algebra.com/algebra/homework/word/geometry/Two-circles-tangent-externally-and-touching-a-given-straight-line.lesson>Two circles tangent externally and touching a given straight line</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=https://www.algebra.com/algebra/homework/word/geometry/A-problem-on-a-circle-touching-another-circle-internally.lesson>A problem on a circle touching another circle internally</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=https://www.algebra.com/algebra/homework/word/geometry/Three-circles-touching-externally-.lesson>Three circles touching externally</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=https://www.algebra.com/algebra/homework/word/geometry/Two-parallel-chords-in-the-intersecting-circles.lesson>Two parallel chords in intersecting circles</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=https://www.algebra.com/algebra/homework/word/geometry/Finding-the-distanse-from-a-point-in-3D-to-a-plane.lesson>Finding the distanse from a point in 3D to a plane</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=https://www.algebra.com/algebra/homework/word/geometry/Geometric-solution-to-one-minimax-problem.lesson>Geometric solution to one minimax problem</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=https://www.algebra.com/algebra/homework/word/geometry/Solving-some-minimax-Geometry-problems.lesson>Solving some minimax Geometry problems</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=https://www.algebra.com/algebra/homework/word/geometry/Advanced-problems-on-finding-area-of-right-angled-triangles.lesson>Advanced problems on finding area of right-angled triangles</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=https://www.algebra.com/algebra/homework/word/geometry/Finding-the-common-shared-area-of-two-triangles.lesson>Finding the common/shared area of two triangles</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=https://www.algebra.com/algebra/homework/word/geometry/Determine-the-type-of-a-triangle.lesson>Determine the type of a triangle</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=https://www.algebra.com/algebra/homework/word/geometry/Selected-problems-on-triangles-similarity.lesson>Selected problems on triangles similarity</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=https://www.algebra.com/algebra/homework/word/geometry/pyramidal-frustum-volume.lesson>Finding pyramidal frustum volume</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=https://www.algebra.com/algebra/homework/word/geometry/Packing-bottles-into-a-box.lesson>Packing bottles into a box</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=https://www.algebra.com/algebra/homework/word/geometry/On-non-existing-a-solution-to-one-trisection-angle-problem.lesson>On non-existing a solution to one trisection angle problem</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=https://www.algebra.com/algebra/homework/word/geometry/Find-a-triangle-with-integer-side-lengths-and-integer-area-.lesson>Find a triangle with integer side lengths and integer area</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=https://www.algebra.com/algebra/homework/word/geometry/Nice-entertainment-problem-on-Area.lesson>Nice entertainment Geometry problems</A>

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<H3>List of lessons with short annotations</H3>

<A HREF=https://www.algebra.com/algebra/homework/word/geometry/Triangle-with-the-side-ratio-456-has-a-smallest-angle-half-of-the-biggest-angle.lesson>Triangle with the sides ratio 4:5:6 has the smallest angle measure half of the biggest angle</A>

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Problem 1. &nbsp;Prove that in any triangle that has the sides proportional with 4:5:6, the smallest angle measure is half of the biggest angle measure.

<A HREF=https://www.algebra.com/algebra/homework/word/geometry/Advanced-problem-on-equilateral-triangles-built-externally-on-sides-of-a-triangle-.lesson>Advanced problem on equilateral triangles built externally on sides of an arbitrary triangle</A>

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Problem 1. &nbsp;Let ABC be any triangle. Equilateral triangles BCX, ACY, and BAZ are constructed such that none of these triangles overlaps triangle ABC.

<A HREF=https://www.algebra.com/algebra/homework/word/geometry/Advanced-problem-on-squares-built-externally-on-sides-of-a-triangle.lesson>Advanced problem on squares built externally on sides of an arbitrary triangle</A>

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Problem 1. &nbsp;Let &nbsp;ABC &nbsp;be a triangle. &nbsp;The squares &nbsp;ABST &nbsp;and &nbsp;ACUV &nbsp;with centers O1 and O2, &nbsp;respectively, &nbsp;are built on its sides as shown in &nbsp;Figure 1.

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(a) &nbsp;Prove that the segments &nbsp;BV&nbsp; and &nbsp;CT&nbsp; are congruent and perpendicular.

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(b) &nbsp;Let &nbsp;M&nbsp; be the midpoint of &nbsp;BC &nbsp;(Figure 2). &nbsp;Prove that the segments &nbsp;MO1&nbsp; and &nbsp;MO2&nbsp; are congruent and perpendicular.

<A HREF=https://www.algebra.com/algebra/homework/word/geometry/Selected-problems-from-the-archive-on-the-area-of-plane-shapes.lesson>Selected problems from the archive on the area of plane shapes</A>

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Problem 1. &nbsp;ABCD &nbsp;is a rectangle with &nbsp;M&nbsp; the midpoint of &nbsp;BC. &nbsp;AC&nbsp; intersects &nbsp;MD&nbsp; at &nbsp;N. &nbsp;Find area of triangle &nbsp;NCD&nbsp; and area of quadrilateral &nbsp;ABMN.

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Problem 2. &nbsp;Two perpendicular chords divide a circle with a radius of &nbsp;13 cm &nbsp;into four parts.

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;If the perpendicular distances of both chords are &nbsp;5 cm &nbsp;each from the center of the circle, &nbsp;find the area of the smallest part.

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Problem 3. &nbsp;A side of an equilateral triangle is the diameter of a semi-circle. &nbsp;If the radius of the semi-circle is 1,

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Problem 4. &nbsp;Trapezoid &nbsp;ABCD &nbsp;has the bases &nbsp;AB &nbsp;(the longer base) &nbsp;and &nbsp;CD &nbsp;(the shorter base).

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;The trapezoid is divided into four triangles by its diagonals &nbsp;AC &nbsp;and &nbsp;BD, &nbsp;intersecting at point &nbsp;O.

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;The areas of two triangles &nbsp;CDO &nbsp;and &nbsp;ABO &nbsp;are &nbsp;9 &nbsp;and &nbsp;25 &nbsp;square units. &nbsp;What is the area of the whole trapezoid?

<A HREF=https://www.algebra.com/algebra/homework/word/geometry/Two-unit-squares-sharing-the-same-center-but-turned-%28rotated%29-each-relative-other.lesson>Two unit squares sharing the same center but turned (rotated) each relative the other</A>

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Problem 1. &nbsp;Two unit squares share the same center. &nbsp;The overlapping region of the two squares is an octagon with perimeter &nbsp;3.5 &nbsp;units.

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;What is the area of the octagon?

<A HREF=https://www.algebra.com/algebra/homework/word/geometry/Finding-the-hypotenuse-of-a-right-angled-triangle-via-its-two-medians.lesson>Finding the hypotenuse of a right-angled triangle via its two medians</A>

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Problem 1. &nbsp;The medians of a right triangle that are drawn from vertices of the acute angles have lengths of &nbsp;{{{2*sqrt(13)}}} &nbsp;and &nbsp;{{{sqrt(73)}}}.

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Find the length of the hypotenuse.

<A HREF=https://www.algebra.com/algebra/homework/word/geometry/Area-of-a-triangle-obtained-by-cutting-uniform-strips-from-a-given-triangle.lesson>Area of a triangle obtained by cutting uniform strips from the given triangle</A>

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Problem 1. &nbsp;A triangle with sides of &nbsp;25 cm, &nbsp;52 cm &nbsp;and &nbsp;63 cm &nbsp;has a second triangle drawn inside it with all its sides at a distance

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;of &nbsp;3 cm &nbsp;from the sides of the original triangle. &nbsp;Find the area of the inner triangle.

<A HREF=https://www.algebra.com/algebra/homework/word/geometry/Perimeter-of-a-thriangle-obtained-by-adding-uniform-strip-to-a-given-triangle.lesson>Find the perimeter of a triangle obtained by adding uniform strip to a given triangle</A>

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Problem 1. &nbsp;Find the perimeter of a triangle obtained from the given triangle with sides &nbsp;7 m, &nbsp;8 m &nbsp;and &nbsp;10 m &nbsp;by adding the strip around its perimeter

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;of uniform width of &nbsp;1 m.

<A HREF=https://www.algebra.com/algebra/homework/word/geometry/Center-of-the-given-circle-is-the-incenter-of-the-given-triangle.lesson>Center of the given circle is the incenter of the given triangle</A>

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Problem 1. &nbsp;A circle is drawn that intersects all three sides of triangle &nbsp;PQR&nbsp; as shown below. &nbsp;Prove that if &nbsp;AB = CD = EF,

<A HREF=https://www.algebra.com/algebra/homework/word/geometry/Determine-the-standard-form-eqn-of-the-circle-inscribed-in-a-triangle.lesson>Determine the standard form equation of the circle inscribed in a triangle</A>

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Problem 1. &nbsp;Determine the standard form equation of the circle inscribed in a triangle,

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;if the triangle has its sides on the lines &nbsp;y=0, &nbsp;4x+3y-50=0, &nbsp;and &nbsp;3x-4y=0. &nbsp;Sketch the graph.

<A HREF=https://www.algebra.com/algebra/homework/word/geometry/Find-the-side-of-a-square-if-distances-are-given-from-an-interior-point-to-3-its-vertices.lesson>Find the side of a square if distances are given from an interior point to 3 its vertices</A>

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Problem 1. &nbsp;From the point inside a square, &nbsp;the distance to three corners are &nbsp;4, &nbsp;5 &nbsp;and &nbsp;6 m, &nbsp;respectively.

<A HREF=https://www.algebra.com/algebra/homework/word/geometry/The-point-which-minimizes-the-sum-of-distances-to-vertices-of-a-given-quadrilateral.lesson>The point which minimizes the sum of distances to vertices of a given quadrilateral</A>

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Problem 1. &nbsp;Find the smallest possible value {{{sqrt(x^2+y^2)}}} + {{{sqrt((x-1)^2+y^2)}}} + {{{sqrt(x^2+(y-1)^2)}}} + {{{sqrt((x-3)^2+(y-4)^2)}}}.

<A HREF=https://www.algebra.com/algebra/homework/word/geometry/Finding-surface-area-of-the-rectangular-box.lesson>Problems on surface area of a rectangular box</A>

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Problem 1. &nbsp;The sum of lengths of all edges of a rectangular box is &nbsp;140 cm &nbsp;and the distance from one corner

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;of the box to the farthest corner is &nbsp;21 cm. &nbsp;What is the total surface area of the box?

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Problem 2. &nbsp;A rectangular box (a PRISM) has the surface area of &nbsp;288 sq. cm.

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;The longest &nbsp;(3D)&nbsp; diagonal of the box is &nbsp;12 cm &nbsp;long. &nbsp;Prove that the box is a cube.

<A HREF=https://www.algebra.com/algebra/homework/word/geometry/Find-the-volume-of-a-rectangular-box-if-the-areas-of-its-faces-are-given.lesson>Find the volume and the dimensions of a rectangular box if the areas of its faces are given</A>

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Problem 1. &nbsp;Find the volume of a rectangular box if three of its faces have areas of &nbsp;30, &nbsp;70 &nbsp;and &nbsp;84 square units.

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Problem 2. &nbsp;Find the dimensions of a rectangular box if three of its faces have areas of &nbsp;30, &nbsp;70 &nbsp;and &nbsp;84 square units.

<A HREF=https://www.algebra.com/algebra/homework/word/geometry/Two-circles-tangent-externally-and-touching-a-given-straight-line.lesson>Two circles tangent externally and touching a given straight line</A>

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Problem 1. &nbsp;If two circles, &nbsp;tangent externally at &nbsp;P, &nbsp;touch a given line at points &nbsp;A&nbsp; and &nbsp;B, &nbsp;prove that angle &nbsp;BPA&nbsp; is a right angle.

<A HREF=https://www.algebra.com/algebra/homework/word/geometry/A-problem-on-a-circle-touching-another-circle-internally.lesson>A problem on a circle touching another circle internally</A>

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Problem 1. &nbsp;Find the center of a circle passing through the points &nbsp;(0,0), &nbsp;(1,0) &nbsp;and touching the circle &nbsp;{{{x^2+y^2}}} = {{{9}}}.

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Problem 2. &nbsp;Find the center of a circle passing through the points &nbsp;(1,0), &nbsp;(2,0) &nbsp;and touching the circle &nbsp;{{{x^2+y^2}}} = {{{9}}}.

<A HREF=https://www.algebra.com/algebra/homework/word/geometry/Three-circles-touching-externally-.lesson>Three circles touching externally</A>

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Problem 1. &nbsp;Two circles with centers &nbsp;A&nbsp; and &nbsp;B&nbsp; are touching externally.

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Third circle with center &nbsp;C&nbsp; touches both the circles &nbsp;A&nbsp; and &nbsp;B&nbsp; externally.

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Suppose &nbsp;AB = 3cm, &nbsp;BC = 3cm &nbsp;and &nbsp;AC = 4cm. &nbsp;Find the radii of the circles.

<A HREF=https://www.algebra.com/algebra/homework/word/geometry/Two-parallel-chords-in-the-intersecting-circles.lesson>Two parallel chords in intersecting circles</A>

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Problem 1. &nbsp;Two circles meet at points &nbsp;X&nbsp; and &nbsp;Y&nbsp;. &nbsp;Line Segment &nbsp;[AXB]&nbsp; meets one circle at &nbsp;A&nbsp; and the other at &nbsp;B.

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Line Segment &nbsp;[CYD]&nbsp; meets one circle at &nbsp;C&nbsp; and the other at &nbsp;D. &nbsp;Prove that &nbsp;AC&nbsp; is parallel to &nbsp;BD.

<A HREF=https://www.algebra.com/algebra/homework/word/geometry/Finding-the-distanse-from-a-point-in-3D-to-a-plane.lesson>Finding the distance from a point in 3D to a plane</A>

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Problem 1. &nbsp;Point &nbsp;P&nbsp; is at &nbsp;10 cm distance from the vertices of a triangle with sides &nbsp;4, &nbsp;5, &nbsp;and &nbsp;6 units long.

<A HREF=https://www.algebra.com/algebra/homework/word/geometry/Geometric-solution-to-one-minimax-problem.lesson>Geometric solution to one minimax problem</A>

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Problem 1. &nbsp;If &nbsp;x+y+z = 7 &nbsp;and &nbsp;xy+yz+zx = 11, &nbsp;then find the least and the largest value of &nbsp;z.

<A HREF=https://www.algebra.com/algebra/homework/word/geometry/Solving-some-minimax-Geometry-problems.lesson>Solving some minimax Geometry problems</A>

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Problem 1. &nbsp;Let &nbsp;(x,y) &nbsp;be a point on the triangular region bounded by the line &nbsp;&nbsp;3x + 4y = 12 &nbsp;&nbsp;and the coordinate axes.

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Determine the points &nbsp;(x,y) &nbsp;in this region which give the minimum and maximum sums of distances of the point

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;from the line &nbsp;&nbsp;3x + 4y = 12 &nbsp;&nbsp;and from the coordinate axes.

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Problem 2. &nbsp;Consider the quarter-ellipse &nbsp;&nbsp;x^2/9 + y^2/4 = 1 &nbsp;&nbsp;at the &nbsp;1st quadrant.

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Two points &nbsp;(0,1/2) &nbsp;and &nbsp;(2,0) &nbsp;are fixed on the &nbsp;y- &nbsp;and &nbsp;x-axes, respectively.

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;A third point &nbsp;(x,y) &nbsp;is allowed to move along the quarter-ellipse, &nbsp;forming a triangle with the two other given points.

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Determine the coordinates of the point &nbsp;(x,y) &nbsp;on the curve which will give the triangle with the maximum area.

<A HREF=https://www.algebra.com/algebra/homework/word/geometry/Advanced-problems-on-finding-area-of-right-angled-triangles.lesson>Advanced problems on finding area of right-angled triangles</A>

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Problem 1. &nbsp;The perimeter of triangle &nbsp;ABC&nbsp; is &nbsp;24. &nbsp;M is the midpoint of &nbsp;AB&nbsp; such that &nbsp;AM = BM = MC = 5. &nbsp;Find the area of the triangle.

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Problem 2. &nbsp;The perimeter of a right triangle is &nbsp;60&nbsp; inches and the length of the altitude to the hypotenuse is &nbsp;12 &nbsp;inches.

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Find the area of the triangle.

<A HREF=https://www.algebra.com/algebra/homework/word/geometry/Finding-the-common-shared-area-of-two-triangles.lesson>Finding the common/shared area of two triangles</A>

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Problem 1. &nbsp;Square &nbsp;ABCD &nbsp;has a side length of &nbsp;8&nbsp; feet.

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Point &nbsp;M&nbsp; is in the midpoint of &nbsp;CD&nbsp; to form triangle &nbsp;ADM.

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Point &nbsp;N&nbsp; is &nbsp;5&nbsp; inches from point &nbsp;C&nbsp; and &nbsp;3&nbsp; inches from point &nbsp;B&nbsp; to form triangle &nbsp;NCD.

<A HREF=https://www.algebra.com/algebra/homework/word/geometry/Determine-the-type-of-a-triangle.lesson>Determine the type of a triangle</A>

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Problem 1. &nbsp;Let &nbsp;a, &nbsp;b &nbsp;and &nbsp;c &nbsp;be the sides of a triangle. &nbsp;If &nbsp;&nbsp;{{{a^2 + b^2 + c^2}}} = {{{ab + bc + ca}}}, &nbsp;&nbsp;determine the type of triangle.

<A HREF=https://www.algebra.com/algebra/homework/word/geometry/Selected-problems-on-triangles-similarity.lesson>Selected problems on triangles similarity</A>

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Problem 1. &nbsp;AB&nbsp; is the diameter of a circle. &nbsp;AD &nbsp;and &nbsp;BC &nbsp;are tangents to the circle with &nbsp;AD = 9cm &nbsp;and &nbsp;BC = 16cm.

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;If &nbsp;AC&nbsp; and &nbsp;BD&nbsp; intersect at a point on the circle, &nbsp;find the length &nbsp;AB.

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Problem 2. &nbsp;In the diagram line &nbsp;AD = 2 cm, &nbsp;EF = 1 cm, &nbsp;and parallel segments are indicated.

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;If the total area of the trapezoid is &nbsp;105 cm^2, &nbsp;what is the area, &nbsp;in cm^2, &nbsp;of triangle &nbsp;AMN ?

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Diagram: https://imgur.com/mLTjbMU

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Problem 3. &nbsp;In the diagram below, &nbsp;ABCG &nbsp;is a parallelogram. &nbsp;Line &nbsp;BF=40 cm &nbsp;and &nbsp;Line &nbsp;FE=16 cm. &nbsp;Find the length of &nbsp;Line &nbsp;ED.

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Diagram: https://imgur.com/a/LdjdTqc

<A HREF=https://www.algebra.com/algebra/homework/word/geometry/pyramidal-frustum-volume.lesson>Finding pyramidal frustum volume</A>

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Problem 1. &nbsp;A frustum of pyramid has of square base of length &nbsp;10 cm and a top square of &nbsp;7 cm.

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;The heights of the frustum is &nbsp;6 cm. Calculate the volume of the frustum.

<A HREF=https://www.algebra.com/algebra/homework/word/geometry/Packing-bottles-into-a-box.lesson>Packing bottles into a box</A>

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Problem 1. &nbsp;What is the maximum number of bottles, &nbsp;each of diameter &nbsp;8 cm, &nbsp;that can be packed into a box

<A HREF=https://www.algebra.com/algebra/homework/word/geometry/On-non-existing-a-solution-to-one-trisection-angle-problem.lesson>On non-existing a solution to one trisection angle problem</A>

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Problem 1. &nbsp;Can there exist a triangle &nbsp;ROS &nbsp;in which the trisectors of angle &nbsp;O &nbsp;intersect &nbsp;RS &nbsp;at &nbsp;D &nbsp;and &nbsp;E &nbsp;with &nbsp;RD = 1, &nbsp;DE = 2, &nbsp;and &nbsp;ES = 4 ?

<A HREF=https://www.algebra.com/algebra/homework/word/geometry/Find-a-triangle-with-integer-side-lengths-and-integer-area-.lesson>Find a triangle with integer side lengths and integer area</A>

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Problem 1. &nbsp;The lengths of the sides of a triangle are positive integers.

<A HREF=https://www.algebra.com/algebra/homework/word/geometry/Nice-entertainment-problem-on-Area.lesson>Nice entertainment Geometry problems</A>

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Problem 1. &nbsp;A barber pole consists of a cylinder of radius &nbsp;10 cm on which one is red, &nbsp;one white, &nbsp;and one blue helix,

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;each of the same width are painted. &nbsp;The cylinder is &nbsp;1 m high. &nbsp;If each stripe makes a constant angle of &nbsp;60 degrees

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;with the vertical axis of the cylinder, &nbsp;how much surface area is covered by the red stripe?

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Problem 2. &nbsp;In parallelogram &nbsp;ABCD, &nbsp;AB = 13, &nbsp;AD = 14, &nbsp;and the length of diagonal &nbsp;AC &nbsp;is &nbsp;15.

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Problem 3. &nbsp;A circle is inscribed in a right triangle that has a hypotenuse of &nbsp;182 cm.

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;If the perimeter of the triangle is &nbsp;420 cm, &nbsp;what is the radius of the inscribed circle?

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Problem 4. &nbsp;A triangle with sides of length &nbsp;36 cm, &nbsp;77 cm, &nbsp;and &nbsp;85 cm is inscribed in a circle.

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Inside the triangle a second circle is inscribed. &nbsp;What is the area in square centimeters,

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;concluded between the two circles?

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Problem 5. &nbsp;A &nbsp;5-12-13 &nbsp;triangle is inscribed in a circle, &nbsp;which is inscribed in a larger &nbsp;5-12-13 &nbsp;triangle.

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;What is the ratio of the area of the smaller triangle to the area of the larger triangle?

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Problem 6. &nbsp;Prove that the sum of the three altitudes of a triangle is less than the sum of three sides of a triangle.

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Problem 7. &nbsp;The side lengths of triangle &nbsp;ABC &nbsp;are &nbsp;22, &nbsp;2x, &nbsp;and &nbsp;55.

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;The side lengths of triangle &nbsp;DEF &nbsp;are &nbsp;11, &nbsp;5, &nbsp;and &nbsp;5.5x.

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Problem 8. &nbsp;With two sides of the lengths of &nbsp;10 cm and two sides of the length of &nbsp;5 cm

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;and with a &nbsp;55-degree angle between the &nbsp;10 cm and &nbsp;5 cm sides, &nbsp;how many quadrilaterals do exist?

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Problem 9. &nbsp;Three spherical planets of radius r are on orbits that keep them within viewing distances

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;of one another. &nbsp;At any instant, &nbsp;each planet has a region that cannot be seen from anywhere

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;on the other two planets. &nbsp;What is the total area of the three unseen regions?

To navigate over all topics/lessons of the Online Geometry Textbook use this file/link &nbsp;<A HREF=https://www.algebra.com/algebra/homework/Triangles/GEOMETRY-your-online-textbook.lesson>GEOMETRY - YOUR ONLINE TEXTBOOK</A>.