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T= (BCA')^2+ (BCA')^2+ (ABC')^2, sum of areas, (HOMC 2009 Q9)

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July 25, 2019
geometryarea of a triangleareas

Problem Statement

Give an acute-angled triangle ABCABC with area SS, let points A′,B′,C′A',B',C' be located as follows: A′A' is the point where altitude from AA on BCBC meets the outwards facing semicirle drawn on BXBX as diameter.Points B′,C′B',C' are located similarly. Evaluate the sum T=(T=(area △BCA′)2+(\vartriangle BCA')^2+(area △CAB′)2+(\vartriangle CAB')^2+(area △ABC′)2\vartriangle ABC')^2.
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