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This Lesson (OVERVIEW of lessons on Miscellaneous adavanced Geometry problems) was created by by ikleyn(52781)
  About ikleyn: OVERVIEW of lessons on Miscellaneous advanced Geometry problemsMy lessons on Miscellaneous advanced Geometry problems in this site are List of lessons with short annotationsTriangle with the sides ratio 4:5:6 has the smallest angle measure half of the biggest angle Problem 1. 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. Advanced problem on equilateral triangles built externally on sides of an arbitrary triangle Problem 1. Let ABC be any triangle. Equilateral triangles BCX, ACY, and BAZ are constructed such that none of these triangles overlaps triangle ABC. Show that, regardless of choice of triangle ABC, we always have AX = BY = CZ Advanced problem on squares built externally on sides of an arbitrary triangle Problem 1. Let ABC be a triangle. The squares ABST and ACUV with centers O1 and O2, respectively, are built on its sides as shown in Figure 1. (a) Prove that the segments BV and CT are congruent and perpendicular. (b) Let M be the midpoint of BC (Figure 2). Prove that the segments MO1 and MO2 are congruent and perpendicular. Selected problems from the archive on the area of plane shapes Problem 1. ABCD is a rectangle with M the midpoint of BC. AC intersects MD at N. Find area of triangle NCD and area of quadrilateral ABMN. Problem 2. Two perpendicular chords divide a circle with a radius of 13 cm into four parts. If the perpendicular distances of both chords are 5 cm each from the center of the circle, find the area of the smallest part. Problem 3. A side of an equilateral triangle is the diameter of a semi-circle. If the radius of the semi-circle is 1, find the area that is inside the triangle but outside the semi-circle. Problem 4. Trapezoid ABCD has the bases AB (the longer base) and CD (the shorter base). The trapezoid is divided into four triangles by its diagonals AC and BD, intersecting at point O. The areas of two triangles CDO and ABO are 9 and 25 square units. What is the area of the whole trapezoid? Two unit squares sharing the same center but turned (rotated) each relative the other Problem 1. Two unit squares share the same center. The overlapping region of the two squares is an octagon with perimeter 3.5 units. What is the area of the octagon? Finding the hypotenuse of a right-angled triangle via its two medians Problem 1. The medians of a right triangle that are drawn from vertices of the acute angles have lengths of Find the length of the hypotenuse. Area of a triangle obtained by cutting uniform strips from the given triangle Problem 1. A triangle with sides of 25 cm, 52 cm and 63 cm has a second triangle drawn inside it with all its sides at a distance of 3 cm from the sides of the original triangle. Find the area of the inner triangle. Find the perimeter of a triangle obtained by adding uniform strip to a given triangle Problem 1. Find the perimeter of a triangle obtained from the given triangle with sides 7 m, 8 m and 10 m by adding the strip around its perimeter of uniform width of 1 m. Center of the given circle is the incenter of the given triangle Problem 1. A circle is drawn that intersects all three sides of triangle PQR as shown below. Prove that if AB = CD = EF, then the center of the circle is the incenter of triangle PQR. Determine the standard form equation of the circle inscribed in a triangle Problem 1. Determine the standard form equation of the circle inscribed in a triangle, if the triangle has its sides on the lines y=0, 4x+3y-50=0, and 3x-4y=0. Sketch the graph. Find the side of a square if distances are given from an interior point to 3 its vertices Problem 1. From the point inside a square, the distance to three corners are 4, 5 and 6 m, respectively. Find the length of the side of a square. The point which minimizes the sum of distances to vertices of a given quadrilateral Problem 1. Find the smallest possible value Problems on surface area of a rectangular box Problem 1. The sum of lengths of all edges of a rectangular box is 140 cm and the distance from one corner of the box to the farthest corner is 21 cm. What is the total surface area of the box? Problem 2. A rectangular box (a PRISM) has the surface area of 288 sq. cm. The longest (3D) diagonal of the box is 12 cm long. Prove that the box is a cube. Find the volume and the dimensions of a rectangular box if the areas of its faces are given Problem 1. Find the volume of a rectangular box if three of its faces have areas of 30, 70 and 84 square units. Problem 2. Find the dimensions of a rectangular box if three of its faces have areas of 30, 70 and 84 square units. Two circles tangent externally and touching a given straight line Problem 1. If two circles, tangent externally at P, touch a given line at points A and B, prove that angle BPA is a right angle. A problem on a circle touching another circle internally Problem 1. Find the center of a circle passing through the points (0,0), (1,0) and touching the circle Problem 2. Find the center of a circle passing through the points (1,0), (2,0) and touching the circle Three circles touching externally Problem 1. Two circles with centers A and B are touching externally. Third circle with center C touches both the circles A and B externally. Suppose AB = 3cm, BC = 3cm and AC = 4cm. Find the radii of the circles. Two parallel chords in intersecting circles Problem 1. Two circles meet at points X and Y . Line Segment [AXB] meets one circle at A and the other at B. Line Segment [CYD] meets one circle at C and the other at D. Prove that AC is parallel to BD. Finding the distance from a point in 3D to a plane Problem 1. Point P is at 10 cm distance from the vertices of a triangle with sides 4, 5, and 6 units long. Find the distance from point P to the plane of the triangle. Geometric solution to one minimax problem Problem 1. If x+y+z = 7 and xy+yz+zx = 11, then find the least and the largest value of z. Solving some minimax Geometry problems Problem 1. Let (x,y) be a point on the triangular region bounded by the line 3x + 4y = 12 and the coordinate axes. Determine the points (x,y) in this region which give the minimum and maximum sums of distances of the point from the line 3x + 4y = 12 and from the coordinate axes. Problem 2. Consider the quarter-ellipse x^2/9 + y^2/4 = 1 at the 1st quadrant. Two points (0,1/2) and (2,0) are fixed on the y- and x-axes, respectively. A third point (x,y) is allowed to move along the quarter-ellipse, forming a triangle with the two other given points. Determine the coordinates of the point (x,y) on the curve which will give the triangle with the maximum area. Advanced problems on finding area of right-angled triangles Problem 1. The perimeter of triangle ABC is 24. M is the midpoint of AB such that AM = BM = MC = 5. Find the area of the triangle. Problem 2. The perimeter of a right triangle is 60 inches and the length of the altitude to the hypotenuse is 12 inches. Find the area of the triangle. Finding the common/shared area of two triangles Problem 1. Square ABCD has a side length of 8 feet. Point M is in the midpoint of CD to form triangle ADM. Point N is 5 inches from point C and 3 inches from point B to form triangle NCD. Find the area shared by triangle ADM and NCD. Determine the type of a triangle Problem 1. Let a, b and c be the sides of a triangle. If Selected problems on triangles similarity Problem 1. AB is the diameter of a circle. AD and BC are tangents to the circle with AD = 9cm and BC = 16cm. If AC and BD intersect at a point on the circle, find the length AB. Problem 2. In the diagram line AD = 2 cm, EF = 1 cm, and parallel segments are indicated. If the total area of the trapezoid is 105 cm^2, what is the area, in cm^2, of triangle AMN ? Diagram: https://imgur.com/mLTjbMU Problem 3. In the diagram below, ABCG is a parallelogram. Line BF=40 cm and Line FE=16 cm. Find the length of Line ED. Diagram: https://imgur.com/a/LdjdTqc Finding pyramidal frustum volume Problem 1. A frustum of pyramid has of square base of length 10 cm and a top square of 7 cm. The heights of the frustum is 6 cm. Calculate the volume of the frustum. Packing bottles into a box Problem 1. What is the maximum number of bottles, each of diameter 8 cm, that can be packed into a box with a square base measuring 792 cm by 792 cm? On non-existing a solution to one trisection angle problem Problem 1. Can there exist a triangle ROS in which the trisectors of angle O intersect RS at D and E with RD = 1, DE = 2, and ES = 4 ? Find a triangle with integer side lengths and integer area Problem 1. The lengths of the sides of a triangle are positive integers. One side has length 17 and the perimeter of the triangle is 54. If the area is also an integer, find the length of the longest side. Nice entertainment Geometry problems Problem 1. A barber pole consists of a cylinder of radius 10 cm on which one is red, one white, and one blue helix, each of the same width are painted. The cylinder is 1 m high. If each stripe makes a constant angle of 60 degrees with the vertical axis of the cylinder, how much surface area is covered by the red stripe? Problem 2. In parallelogram ABCD, AB = 13, AD = 14, and the length of diagonal AC is 15. What is the area of the parallelogram ? Problem 3. A circle is inscribed in a right triangle that has a hypotenuse of 182 cm. If the perimeter of the triangle is 420 cm, what is the radius of the inscribed circle? Problem 4. A triangle with sides of length 36 cm, 77 cm, and 85 cm is inscribed in a circle. Inside the triangle a second circle is inscribed. What is the area in square centimeters, concluded between the two circles? Problem 5. A 5-12-13 triangle is inscribed in a circle, which is inscribed in a larger 5-12-13 triangle. What is the ratio of the area of the smaller triangle to the area of the larger triangle? Problem 6. Prove that the sum of the three altitudes of a triangle is less than the sum of three sides of a triangle. Problem 7. The side lengths of triangle ABC are 22, 2x, and 55. The side lengths of triangle DEF are 11, 5, and 5.5x. Is it possible that the triangles are similar? Problem 8. With two sides of the lengths of 10 cm and two sides of the length of 5 cm and with a 55-degree angle between the 10 cm and 5 cm sides, how many quadrilaterals do exist? Problem 9. Three spherical planets of radius r are on orbits that keep them within viewing distances of one another. At any instant, each planet has a region that cannot be seen from anywhere on the other two planets. What is the total area of the three unseen regions? 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