MathDB

4

Part of 2017 Czech-Polish-Slovak Junior Match

Problems(2)

Czech-Polish-Slovak Junior geometry - excircle related

Source: Czech-Polish-Slovak Junior Match 2017, individual p4 CPSJ

2/20/2020
Given is a right triangle ABCABC with perimeter 22, with ∠B=90o\angle B=90^o . Point SS is the center of the excircle to the side ABAB of the triangle and HH is the intersection of the heights of the triangle ABSABS . Determine the smallest possible length of the segment HSHS .
geometryperimeterminright triangle
reconstruct a trapezoid given two segments

Source: Czech-Polish-Slovak Junior Match 2017, Team p4 CPSJ

3/6/2020
Bolek draw a trapezoid ABCDABCD trapezoid (AB//CDAB // CD) on the board, with its midsegment line EFEF in it. Point intersection of his diagonal AC,BDAC, BD denote by P,P, and his rectangular projection on line ABAB denote by QQ. Lolek, wanting to tease Bolek, blotted from the board everything except segments EFEF and PQPQ. When Bolek saw it, wanted to complete the drawing and draw the original trapezoid, but did not know how to do it. Can you help Bolek?
geometrytrapezoidconstruction