MathDB

P4

Part of 2020 Dürer Math Competition (First Round)

Problems(2)

Orthocenters form an equilateral triangle

Source: 2019-20 International Dürer Competition, Category E ,P4

8/19/2020
Let ABCABC be an acute triangle with side ABAB of length 11. Say we reflect the points AA and BB across the midpoints of BCBC and ACAC, respectively to obtain the points A’A’ and B’B’ . Assume that the orthocenters of triangles ABC ABC, A’BCA’BC and B’ACB’AC form an equilateral triangle. a) Prove that triangle ABCABC is isosceles. b) What is the length of the altitude of ABCABC through CC?
geometryEquilateral Triangle
Construction of a triangle

Source: 2019-20 International's Dürer Competition, Category E+,P4

8/19/2020
Suppose that you are given the foot of the altitude from vertex AA of a scalene triangle ABCABC, the midpoint of the arc with endpoints BB and CC, not containing AA of the circumscribed circle of ABCABC, and also a third point PP. Construct the triangle from these three points if PP is the a) orthocenter b) centroid c) incenter of the triangle.
geometryConstruct