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Why are the last 3 all geo...

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April 26, 2022

Problem Statement

Let PP, AA, BB, CC, DD be points on a plane such that PA=9PA = 9, PB=19PB = 19, PC=9PC = 9, PD=5PD = 5, ∠APB=120∘\angle APB = 120^\circ, ∠BPC=45∘\angle BPC = 45^\circ, ∠CPD=60∘\angle CPD = 60^\circ, and ∠DPA=135∘\angle DPA = 135^\circ. Let G1G_1, G2G_2, G3G_3, and G4G_4 be the centroids of triangles PABPAB, PBCPBC, PCDPCD, PDAPDA. [G1G2G3G4][G_1G_2G_3G_4] can be expressed as ab+cda\sqrt{b} + c\sqrt{d}. Find a+b+c+da+b+c+d.
2022 CCA Math Bonanza Individual Round #15
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