Geometry, Plane Geometry, Triangles
This subject explores the properties, types (e.g., equilateral, isosceles, scalene, right-angled), and theorems related to triangles, which are fundamental three-sided polygons. Questions often involve calculating angles, side lengths, area, perimeter, and applying triangle-specific theorems.
Triangle Congruence Triangle Similarity-
Right Triangles and a Square
Given a large number of congruent right triangles.
The side lengths of each triangle are 3, 4, and 5.
What is the maximum number of such triangles that can be placed inside a 20×20 square, such that they do not overlap?
Sources:Topics:Geometry -> Area Calculation Geometry -> Plane Geometry -> Triangles Geometry -> Plane Geometry -> Pythagorean Theorem- Gillis Mathematical Olympiad, 2016-2017 Question 3
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Hexagon and Triangle
A regular hexagon and an equilateral triangle have the same perimeter. The area of the triangle is known to be 60. Find the area of the hexagon.
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Orange Star of David
The area of the blue triangle is equal to 1. Calculate the area of the orange Star of David:
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Angles
Calculate the sum of the marked angles:
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How many triangles?
How many triangles are there in the picture?
Sources:Topics:Geometry -> Plane Geometry -> Triangles Combinatorics -> Case Analysis / Checking Cases -> Processes / Procedures Geometry -> Plane Geometry -> Angle Calculation Number Theory -> Division -
How Many Triangles - 2?
How many triangles are in the picture?
Sources:Topics:Geometry -> Area Calculation Geometry -> Plane Geometry -> Triangles Combinatorics -> Case Analysis / Checking Cases -> Processes / Procedures -
Length of the Segment
On the side BC of triangle ABC, a point D is marked. The perimeter of triangle ABC is 15 centimeters, the perimeter of triangle ABD is 12 centimeters, and the perimeter of triangle ACD is 13 centimeters.
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What is the length of segment AD? -
Inscribed Circle in a Triangle
Inside a triangle there is a point P, whose distances from the lines containing the sides of the triangle are `d_a,d_b,d_c`. Let R denote the radius of the circumscribed circle of the triangle and r the radius of the inscribed circle in the triangle. Show that `sqrt(d_a)+sqrt(d_b)+sqrt(d_3)<= sqrt (2R+5r) `.
Sources:Topics:Geometry -> Plane Geometry -> Triangles Geometry -> Plane Geometry -> Circles Algebra -> Inequalities- Gillis Mathematical Olympiad, 2019-2020 Question 7
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Triangle Side Lengths
Let `n > 2` be an integer, and let ` t_1,t_2,...,t_n` be positive real numbers such that
`(t_1+t_2+...+t_n)(1/t_1 + 1/t_2 + ... + 1/t_n) < n^2+1`
Prove that for all i,j,k such that `1<=i<j<k<=n`, the triple of numbers `t_i,t_j,t_k` are the side lengths of a triangle.
Sources:Topics:Geometry -> Plane Geometry -> Triangles Algebra -> Inequalities Proof and Example -> Proof by Contradiction Geometry -> Plane Geometry -> Triangle Inequality- Grossman Math Olympiad, 2006 Question 5
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Paths in a Triangular Park
In a park, there are 3 straight paths that form a triangle (there are no additional paths). The entrances to the park are at the midpoints of the paths, and a lamp hangs at each vertex of the triangle. From each entrance, the shortest walking distance along the park's paths to the lamp at the opposite vertex was measured. It turned out that 2 out of the 3 distances are equal to each other. Is the triangle necessarily isosceles?
Sources:Topics:Geometry -> Plane Geometry -> Triangles Proof and Example -> Constructing an Example / Counterexample Geometry -> Plane Geometry -> Triangle Inequality- Beno Arbel Olympiad, 2017, Grade 8 Question 3