Lesson OVERVIEW of my additional lessons on miscellaneous advanced Geometry problems

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<H2>OVERVIEW of my additional lessons on miscellaneous advanced Geometry problems</H2>

My additional lessons on miscellaneous advanced Geometry problems in this site are

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=https://www.algebra.com/algebra/homework/word/geometry/Find-the-rate-of-moving-of-the-tip-of-a-shadow.lesson>Find the rate of moving of the tip of a shadow</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=https://www.algebra.com/algebra/homework/word/geometry/A-radio-transmitter-accessibility-area.lesson>A radio transmitter accessibility area</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=https://www.algebra.com/algebra/homework/word/geometry/Miscellaneous-geometric-problems.lesson>Miscellaneous geometric problems</A>

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

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=https://www.algebra.com/algebra/homework/word/geometry/Remarkable-properties-of-triangles-into-which-diagonals-divide-a-quadrilateral.lesson>Remarkable properties of triangles into which diagonals divide a quadrilateral</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=https://www.algebra.com/algebra/homework/word/geometry/A-trapezoid-divided-in-four-triangles-by-its-diagonals.lesson>A trapezoid divided in four triangles by its diagonals</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=https://www.algebra.com/algebra/homework/word/misc/A-problem-on-heptagon.lesson>A problem on a regular heptagon</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=https://www.algebra.com/algebra/homework/word/geometry/The-area-of-a-regular-octagon.lesson>The area of a regular octagon</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=https://www.algebra.com/algebra/homework/word/geometry/The-fraction-of-the-area-of-a-regular-octagon.lesson>The fraction of the area of a regular octagon</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=https://www.algebra.com/algebra/homework/word/geometry/Try-to-solve-this-nice-Geometry-problem.lesson>Try to solve these nice Geometry problems !</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=https://www.algebra.com/algebra/homework/word/geometry/Find-the-angle-between-sides-of-folded-triangle.lesson>Find the angle between sides of folded triangle</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=https://www.algebra.com/algebra/homework/word/geometry/A-problems-on-three-spheres.lesson>A problem on three spheres</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=https://www.algebra.com/algebra/homework/word/geometry/A-sphere-placed-in-an-inverted-cone.lesson>A sphere placed in an inverted cone</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=https://www.algebra.com/algebra/homework/word/geometry/07-An-upper-level-Geometry-problem-on-special-%2815-30-135%29-triangle.lesson>An upper level Geometry problem on special (15°,30°,135°)-triangle</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=https://www.algebra.com/algebra/homework/word/geometry/A-great-Math-Olympiad-level-Geometry-problem.lesson>A great Math Olympiad level Geometry problem</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=https://www.algebra.com/algebra/homework/word/geometry/Nice-geometry-problem-of-a-Math-Olympiad-level.lesson>Nice geometry problem of a Math Olympiad level</A>

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

<A HREF=https://www.algebra.com/algebra/homework/word/geometry/Find-the-rate-of-moving-of-the-tip-of-a-shadow.lesson>Find the rate of moving of the tip of a shadow</A>

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Problem 1. &nbsp;A street light is at the top of a 18 ft tall pole. &nbsp;A man &nbsp;6 &nbsp;ft tall walks away

&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 pole with a speed of &nbsp;8 &nbsp;ft/sec along a straight path.

&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;How fast is the tip a shadow moving when he is &nbsp;40 &nbsp;ft from the base of the pole?

<A HREF=https://www.algebra.com/algebra/homework/word/geometry/A-radio-transmitter-accessibility-area.lesson>A radio transmitter accessibility area</A>

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Problem 1. &nbsp;A small radio transmitter broadcasts in a 21 mile radius. If you drive along a straight line

&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 a city 25 miles north of the transmitter to a second city 29 miles east of the transmitter,

&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;during how much of the drive will you pick up a signal from the transmitter?

<A HREF=https://www.algebra.com/algebra/homework/word/geometry/Miscellaneous-geometric-problems.lesson>Miscellaneous geometric problems</A>

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Problem 1. &nbsp;What is the distance between the tips of the minute hand and the hour hand

&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 a clock at &nbsp;1:35 pm, &nbsp;where the length of the minute hand is &nbsp;14 cm

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Problem 2. &nbsp;Two circles with the centers at &nbsp;A &nbsp;and &nbsp;B &nbsp;intersect and share a common chord &nbsp;CD.

&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 radius of circle &nbsp;A &nbsp;is &nbsp;10 in, &nbsp;the radius of circle &nbsp;B &nbsp;is &nbsp;16 &nbsp;in. &nbsp;The distance

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Problem 3. &nbsp;A paper cone of circular base has the diameter of &nbsp;10 cm and the height of&nbsp;12 cm. It is flattened out

&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;into the sector of a circle. &nbsp;What is the angle of the sector?

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Problem 4. &nbsp;Gibb’s &nbsp;Hill &nbsp;Lighthouse, &nbsp;Southampton, &nbsp;Bermuda, &nbsp;in operation since &nbsp;1846, &nbsp;stands &nbsp;117 &nbsp;feet high

&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 a hill &nbsp;245 &nbsp;feet high, &nbsp;so its beam of light is &nbsp;362 &nbsp;feet above sea level.

&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 distance from the top of the lighthouse to the horizon.

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Problem 5. &nbsp;Regular hexagon &nbsp;ABCDEF &nbsp;has sides of length &nbsp;2. &nbsp;The point &nbsp;P &nbsp;is the midpoint of &nbsp;AB. &nbsp;Q &nbsp;is the midpoint of &nbsp;BC &nbsp;and so on.

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Problem 6. &nbsp;An archery target is constructed of five concentric circles such that the area of the inner 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;is equal to the area of each of the four rings. &nbsp;If the radius of the outer circle is &nbsp;12 m,

&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 width of the band between the second and the third circle.

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Problem 7. &nbsp;The sum of the distance from a point &nbsp;P &nbsp;to &nbsp;(4,0) &nbsp;and &nbsp;(-4,0) &nbsp;is &nbsp;9.

&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 abscissa of &nbsp;P &nbsp;is &nbsp;1, &nbsp;find its ordinate.

<A HREF=https://www.algebra.com/algebra/homework/word/geometry/Miscellaneous-problems-on-parallelograms.lesson>Miscellaneous problems on parallelograms</A>

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Problem 1. &nbsp;The diagonals of a parallelogram measure &nbsp;16 cm and &nbsp;24 cm. &nbsp;The shorter side measures &nbsp;10 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;&nbsp;&nbsp;&nbsp;&nbsp;(a)  &nbsp;&nbsp;Find the area of the parallelogram.

&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;&nbsp;&nbsp;&nbsp;&nbsp;(b)  &nbsp;&nbsp;Find the measure of the longer side.

&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;&nbsp;&nbsp;&nbsp;&nbsp;(c)  &nbsp;&nbsp;Find the measure of the smaller angle of the parallelogram.

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Problem 2. &nbsp;A parallelogram has diagonals &nbsp;34 in and &nbsp;20 in and one side measures &nbsp;15 in.

&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;&nbsp;&nbsp;&nbsp;&nbsp;(a)  &nbsp;&nbsp;Find the length of the other side.

&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;&nbsp;&nbsp;&nbsp;&nbsp;(b)  &nbsp;&nbsp;Find the area of the parallelogram.

&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;&nbsp;&nbsp;&nbsp;&nbsp;(c)  &nbsp;&nbsp;Find the largest interior angle of the parallelogram.

<A HREF=https://www.algebra.com/algebra/homework/word/geometry/Remarkable-properties-of-triangles-into-which-diagonals-divide-a-quadrilateral.lesson>Remarkable properties of triangles into which diagonals divide a quadrilateral</A>

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Problem 1. &nbsp;Let &nbsp;ABCD &nbsp;is a convex quadrilateral and &nbsp;AC &nbsp;and &nbsp;BD &nbsp;are its diagonals

&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 quadrilateral is a parallelogram, &nbsp;then triangles &nbsp;APB, &nbsp;BPC, &nbsp;CPD &nbsp;and &nbsp;DPA &nbsp;all have the same area.

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Problem 2. &nbsp;Let &nbsp;ABCD &nbsp;is an arbitrary convex quadrilateral and &nbsp;AC &nbsp;and &nbsp;BD &nbsp;are its diagonals

&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 triangles &nbsp;APB, &nbsp;BPC, &nbsp;CPD &nbsp;and &nbsp;DPA &nbsp;all have the same area, &nbsp;then quadrilateral &nbsp;ABCD &nbsp;is a parallelogram.

<A HREF=https://www.algebra.com/algebra/homework/word/geometry/A-trapezoid-divided-in-four-triangles-by-its-diagonals.lesson>A trapezoid divided in four triangles by its diagonals</A>

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Problem 1. &nbsp;ABCD &nbsp;is a trapezoid in which &nbsp;AD:BC = 3:5. &nbsp;If the area of

&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;triangle &nbsp;AMD &nbsp;is &nbsp;315 cm^2, &nbsp;find the area of the trapezoid.

<A HREF=https://www.algebra.com/algebra/homework/word/misc/A-problem-on-heptagon.lesson>A problem on a regular heptagon</A>

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Problem 1. &nbsp;Given regular heptagon &nbsp;ABCDEFG, &nbsp;a circle can be drawn that is tangent to &nbsp;DC &nbsp;at &nbsp;C &nbsp;and &nbsp;to &nbsp;EF &nbsp;at &nbsp;F.

&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 radius of the circle if the side length of the heptagon is &nbsp;1?

<A HREF=https://www.algebra.com/algebra/homework/word/geometry/The-area-of-a-regular-octagon.lesson>The area of a regular octagon</A>

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Problem 1. &nbsp;We cut a regular octagon &nbsp;ABCDEFGH &nbsp;out of a piece of cardboard.

<A HREF=https://www.algebra.com/algebra/homework/word/geometry/The-fraction-of-the-area-of-a-regular-octagon.lesson>The fraction of the area of a regular octagon</A>

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Problem 1. &nbsp;Vertices of a regular octagon are connected as shown in the attached &nbsp;Figure. &nbsp;What fraction of the octagon area is shaded?

<A HREF=https://www.algebra.com/algebra/homework/word/geometry/Try-to-solve-this-nice-Geometry-problem.lesson>Try to solve these nice Geometry problems !</A>

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Problem 1. &nbsp;Two sides of triangle &nbsp;ABC &nbsp;are &nbsp;AB = 34 cm &nbsp;and &nbsp;AC = 25 cm and their included angle measures &nbsp;62°.

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Problem 2. &nbsp;The circle circumscribes an equilateral triangle. &nbsp;The area of the circle is &nbsp;12pi.

&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 triangle's perimeter ?

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Problem 3. &nbsp;The center of a circle is at &nbsp;(-3,-2). &nbsp;If a chord of length &nbsp;4 &nbsp;is bisected at &nbsp;(3,1), &nbsp;find the length of the radius.

<A HREF=https://www.algebra.com/algebra/homework/word/geometry/Find-the-angle-between-sides-of-folded-triangle.lesson>Find the angle between sides of folded triangle</A>

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Problem 1. &nbsp;An isosceles triangle &nbsp;ABC, &nbsp;in which &nbsp;AB = BC = {{{6*sqrt(2)}}} &nbsp;and &nbsp;AC = 12

&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;is folded along the altitude &nbsp;BD, &nbsp;so that planes &nbsp;ABD &nbsp;and &nbsp;BDC &nbsp;form

&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 right dihedral angle. &nbsp;Find the angle between side &nbsp;AB &nbsp;and its new position.

<A HREF=https://www.algebra.com/algebra/homework/word/geometry/A-problems-on-three-spheres.lesson>A problem on three spheres</A>

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Problem 1. &nbsp;The center of each of three spheres of radius &nbsp;R &nbsp;lies in the surfaces

&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 other two. &nbsp;Pass a plane containing the centers of the spheres.

&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 common to the three great circles cut from the spheres by this plane.

<A HREF=https://www.algebra.com/algebra/homework/word/geometry/A-sphere-placed-in-an-inverted-cone.lesson>A sphere placed in an inverted cone</A>

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Problem 1. &nbsp;A sphere is placed in an inverted hollow conical vessel of base radius &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 vertical height &nbsp;12 cm. &nbsp;If the highest point of the sphere is at the level of the base

<A HREF=https://www.algebra.com/algebra/homework/word/geometry/07-An-upper-level-Geometry-problem-on-special-%2815-30-135%29-triangle.lesson>An upper level Geometry problem on special (15°,30°,135°)-triangle</A>

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Problem 1. &nbsp;ABC &nbsp;is a triangle with &nbsp;∠CAB=15° &nbsp;and &nbsp;∠ABC=30°.

&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;M &nbsp;is the midpoint of &nbsp;AB, &nbsp;find &nbsp;∠ACM.

<A HREF=https://www.algebra.com/algebra/homework/word/geometry/A-great-Math-Olympiad-level-Geometry-problem.lesson>A great Math Olympiad level Geometry problem</A>

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Problem 1. &nbsp;In triangle &nbsp;ABC, &nbsp;point &nbsp;X &nbsp;is on side &nbsp;BC &nbsp;such that &nbsp;AX = 13, &nbsp;BX = 13, &nbsp;CX = 5,

&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 the circumcircles of triangles &nbsp;ABX &nbsp;and &nbsp;ACX &nbsp;have the same radius. &nbsp;Find the area of triangle &nbsp;ABC.

<A HREF=https://www.algebra.com/algebra/homework/word/geometry/Nice-geometry-problem-of-a-Math-Olympiad-level.lesson>Nice geometry problem of a Math Olympiad level</A>

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Problem 1. &nbsp;The area of &nbsp;△ABC &nbsp;is &nbsp;40. &nbsp;Points &nbsp;P, &nbsp;Q &nbsp;and &nbsp;R &nbsp;lie on sides &nbsp;AB, &nbsp;BC &nbsp;and &nbsp;CA &nbsp;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;If &nbsp;AP = 3 &nbsp;and &nbsp;PB = 5, &nbsp;and the area of &nbsp;△ABQ &nbsp;is equal to the area of &nbsp;PBQR, &nbsp;determine the area of &nbsp;△AQC.

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>.