MathLabs

Problem 1

Let ABCABC be an acute triangle inscribed in a circle Γ\Gamma. Let A1A_1 be the orthogonal projection of AA onto BCBC, so that AA1AA_1 is an altitude. Let B1B_1 and C1C_1 be the orthogonal projections of A1A_1 onto ABAB and ACAC, respectively. Point PP is such that quadrilateral AB1PC1AB_1PC_1 is convex and has the same area as triangle ABCABC. Is it possible that PP lies strictly in the interior of circle Γ\Gamma? Justify your answer.
Step 1 of 5: Triangle AB1C1 is similar to ABC
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AC1=AA1sin⁡C,AB1=AA1sin⁡B,B1C1=AA1sin⁡AAC_1=AA_1\sin C,\quad AB_1=AA_1\sin B,\quad B_1C_1=AA_1\sin A
Detailed analysis

Since ∠AA1B1=∠AA1C1=90∘\angle AA_1B_1=\angle AA_1C_1=90^\circ, points B1,C1B_1,C_1 lie on the circle with diameter AA1AA_1. In that circle, AC1=AA1sin⁡∠AA1C1=AA1sin⁡(90∘−∠A1AC)=AA1sin⁡CAC_1=AA_1\sin\angle AA_1C_1=AA_1\sin(90^\circ-\angle A_1AC)=AA_1\sin C, and likewise AB1=AA1sin⁡BAB_1=AA_1\sin B and B1C1=AA1sin⁡AB_1C_1=AA_1\sin A; hence triangle AC1B1AC_1B_1 is similar to triangle ABCABC.