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Welcome to Geometry Proposed Problems!
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maintained by Antonio Gutierrez.
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Proposed Problems - Table of Content: Level: High
School, SAT Prep, College geometry.
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Proposed Problem
107.Angles, Triangle. Cevian.  |
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Proposed Problem
106.Angles, Triangle. Cevian. |
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Proposed Problem
105.Angles, Triangle. Interior
Point. |
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Proposed Problem
104.Angles, Triangle. |
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Proposed Problem
103.Equilateral Triangle Area,
Interior Point, Heron's Formula. |
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Proposed Problem
102.Equilateral Triangle Area,
Interior Point. |
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Proposed Problem
101.Equilateral Triangle,
Interior Point, Pythagorean Theorem, Angles. |
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Proposed Problem
100. Circle Area, Archimedes' Book of Lemmas. |
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Proposed Problem 99: Circle Area, General Extension to Pythagoras' Theorem. |
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Proposed Problem
98. Quadrilateral Area, Centroid,
Similarity. |
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Proposed Problem
97. Similar Triangles, Areas. |
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Proposed Problem
96. Similar Triangles, Incenters, Parallelogram. |
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Proposed Problem
95. Similar Triangles, Inradii,
Parallel. |
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Proposed Problem
94. Similar Triangles, Circumcircles,
Circumradii. |
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Proposed Problem
93. Similar Triangles, Circumcircles, Parallelogram. |
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Proposed Problem
92. Similar Triangles, Circumcircles, Circumradii, Parallel. |
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Proposed Problem
91. Similar Triangles, Altitude, Parallel. |
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Proposed Problem
90. Quadrilateral and Triangle
Areas, Midpoints. |
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Proposed Problem
89. Triangle Areas, Midpoints. |
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Proposed Problem
88. Triangle and Quadrilateral Areas, Midpoints. |
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Proposed Problem
87. Area of Quadrilaterals and Midpoints of Diagonals. |
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Proposed Problem
86. Intouch and Extouch Triangles, Areas. |
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Proposed Problem
85. Contact Triangles Areas, Incircle, Excircle. |
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Proposed Problem
84. Contact Triangles Areas, Incircle, Excircle, Inradius, Exradius. |
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Proposed Problem
83. Area of the Excircle Contact
Triangle, exradius, circumradius. |
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Proposed Problem
82: Triangle. Area of the Contact
Triangle,
inradius, circumradius. |
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Proposed Problem
81: Triangle. Area of a triangle,
side, inradius, circumradius. |
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Proposed Problem
80: Triangle. Area of a triangle,
side, incircle, inradius. |
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Proposed Problem
79: Triangle.
Similarity, Altitudes, Orthocenter, Incircles, Inradii. |
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Proposed Problem
78: Angles of a Circle.
Perpendicular and parallel
lines, Midpoint, Diameter, Chord, Cyclic quadrilateral, Congruence. |
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Proposed Problem
77: Angles of a Circle. Parallel
lines, Cyclic quadrilateral. |
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Proposed Problem
76: Area of a Circle. Square,
Circle, Circular Sector. |
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Proposed Problem
75: Three Intersecting Circles. Cyclic
quadrilateral, Angles. |
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Proposed Problem
74: Three Intersecting Circles. Cyclic
quadrilateral, Angles. |
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Proposed Problem
73: Three Intersecting Circles. Cyclic
quadrilateral. |
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Proposed Problem
72: Intersecting Circles. Cyclic
quadrilateral, Chords, Parallel. |
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Proposed Problem
71: Cyclic Quadrilateral. |
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Proposed Problem
70: Squares inscribed in a triangle, Similarity. |
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Proposed Problem
69: Square inscribed in a triangle, Similarity. |
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Proposed Problem
68: Triangle, Incircle, Inradius, Tangent, Similarity. |
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Proposed Problem
67: Triangle, Circumcircle, Angles, Cyclic Quadrilateral. |
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Proposed Problem
66: Triangle, Excircle, Tangents, Geometric Mean. |
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Proposed Problem
65: Right Triangle, Mindpoints. |
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Proposed Problem
64: Triangle, Incircle, Transversal. |
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Proposed Problem
63: Regular Heptagon, Side and Diagonals. |
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Proposed Problem
62: Square Diagonal, Inscribed Circle. |
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Proposed Problem
61: Triangle, Trisection of Sides. |
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Proposed Problem
60: Isosceles triangle. |
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Proposed Problem
59: Right and Equilateral Triangles, Midpoints. |
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Proposed Problem
58: Right Triangle, Congruence, Perpendiculars. |
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Proposed Problem
57: Angle bisector, circles Cyclic
Quadrilateral. |
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Proposed Problem
56: Angle bisector, circles Parallel
Lines. |
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Proposed Problem
55: Angle bisector, circles Cyclic
Quadrilateral. |
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Proposed Problem
54: Angle bisector, circles Midpoint of
arc. |
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Proposed Problem
53: Angle bisector, circles |
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Proposed Problem 52: Triangle, angles, incircle.
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Proposed Problem 51:
Fagnano's Problem
Inscribed Triangle with the Minimum Perimeter. |
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Problem 50. Triangle
with Equilateral triangles. |
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Problem 49. Right triangle, cevian,
perpendicular, and angles. |
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Problem 48. Angles and triangles. |
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Problem 47. Triangle, median, and
angles. |
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Problem 46. Triangle, median, and
angles. |
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Problem 45. Angles and triangles. |
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Problem 44. Angles and triangles. |
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Problem 43. Angles equal segments, and triangles. |
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Problem 42. Angles, equal
segments, and triangles. |
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Sangaku Problem 41. "Mickey
Mouse."
Three circles, tangents, radii. |
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Problem 40.
Triangle, Incenter, Excenter, Angles 80, 40, Distances. |
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Problem 40. Geometry Help.
Suggestions. |
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Problem 39.
Triangle, Incircle, Bisector, Cyclic Quadrilateral and angles. |
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Problem 39 Geometry Help.
Facts you should know for the proposed problem 39. |
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Problem 38.
Right
triangle, altitude, incircles, incenters, and angles. |
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Problem 37.
Right
triangle, altitude, incircles, incenters, and orthocenter. |
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Problem 36.
Right
triangle, altitude, incircles and inradii. |
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Problem 35.
Incenters and Inradii in Cyclic Quadrilateral. |
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Problem 34.
Right
triangle, Cevian, Incircles, Tangents and Inradius. |
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Problem 33.
Triangle and quadrilateral. |
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Problem 32.
Triangle, Cevian, Incircles, Tangents. |
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Problem 31.
Right
Triangle, Incircle, Collinears.
Problem 30.
Right
Triangle, Incircle, Inradius. |
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Problem 29: Geometry Help.
Facts you should know. |
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Problem 29.
Right
Triangle, altitude, incircle and inradius. Ten conclusions. |
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Problem 28.
Right
Triangle, altitude, incircles and inradius. |
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Problem 27.
Right
Triangle, incircles and inradius. |
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Problem 26.
Right
Triangle, altitude, incircles and inradius. |
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Problem 25.
Right
triangle, altitude, incircles and inradius. |
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Problem 24.
Right
triangle, altitude, incircles and inradii. |
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Problem 23.
Right
triangle, altitude, incircles and inradii. |
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Problem 22.
Right
triangle, altitudes, incircles and inradii. |
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Problem 21.
Acute triangle, orthocenter, diameter, tangents. |
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Problem
20.
Right triangle, altitude, incircles and inradii. |
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Problem 19.
Right triangle, perpendicular, and Excenter. |
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Problem 18.
Right triangle, cevian, and angles. |
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Problem 17.
Right triangle, altitude, and angles. |
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Problem 16.
Triangle, cevian, perpendicular, and angles. |
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Problem 15.
Triangle, cevian, sum of segments, and angles. |
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Problem 14.
Triangle, median, and angles. |
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Problem 13.
Triangle, cevian, equal segments, and angles. |
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Problem 12.
Triangle, cevian, equal segments, and angles. |
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Problem 11.
Right triangle, cevian, and angles. |
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Problem 10.
Triangle, cevian, equal segments, and angles. |
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Problem 9.
Triangles, equal segments, and angles. |
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Problem 8.
Triangles, equal segments, and angles. |
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Problem 7.
Triangle, cevian, equal segments, and angles. |
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Problem 6.
Triangle, cevian, equal segments, and angles. |
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Problem 5.
Triangle, cevian, equal segments, and angles. |
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Problem 4.
Quadrilateral, equal sides and angles. |
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Problem 3.
Triangle, median and angles. |
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Problem 2.
Triangle and angles. |
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Geometry Help: Key theorems
or Facts you should know for the proposed problems. |
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Problem 1.
Triangle, median and angles. |
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 "A great discovery solves a great
problem, but there is a grain of discovery in the solution of
any problem. Your problem may be modest, but if it challenges
your curiosity and brings into play your inventive faculties,
and if you solve it by your own means, you may experience the
tension and enjoy the triumph of discovery. Such expert
experiences at a susceptible age may create a taste for mental
work and leave their imprint on mind and character for a
lifetime." George Polya, 1944.
"Four phases trying to find the
solution, we may repeatedly change our point of view, our way of
looking at the problem. We have to shift our position again and
again. Our conception of the problem is likely to be rather
incomplete when we start the work; our outlook look is different
when we have made some progress; it is again different when we
have almost obtained the solution. In order to group
conveniently the questions and suggestions of our list, we shall
distinguish four phases of the work. First we have to
understand the problem; we have to see clearly what is
required. Second, we have to see how the various items
are connected, how the unknown-known is linked to the data in
order to obtain the idea of the solution, to make a plan.
Third, we carry out our plan. Fourth, we look back at
the completed solution, we review and discuss it." George Polya,
1944.
Exercise your brain. Solve these problems about
congruence of line segments, angles, triangles,
circumferences, similarity, and areas and lift up your geometry skills. Designed for high-school students,
college geometry, SAT preparation and teachers with an interest
in geometry problem-solving.
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