MathDB

2020 Caucasus Mathematical Olympiad

Part of Caucasus Mathematical Olympiad

Subcontests

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Two equal common tangents are perpendicular

Let ω1\omega_1 and ω2\omega_2 be two non-intersecting circles. Let one of its internal tangents touches ω1\omega_1 and ω2\omega_2 at A1A_1 and A2A_2, respectively, and let one of its external tangents touches ω1\omega_1 and ω2\omega_2 at B1B_1 and B2B_2, respectively. Prove that if A1B2=A2B1A_1B_2 = A_2B_1, then A1B2⊥A2B1A_1B_2 \perp A_2B_1.

If two common tangents are perpendicular then they are equal

Let ω1\omega_1 and ω2\omega_2 be two non-intersecting circles. Let one of its internal tangents touches ω1\omega_1 and ω2\omega_2 at A1A_1 and A2A_2, respectively, and let one of its external tangents touches ω1\omega_1 and ω2\omega_2 at B1B_1 and B2B_2, respectively. Prove that if A1B2⊥A2B1A_1B_2\perp A_2B_1, then A1B2=A2B1A_1B_2 = A_2B_1.
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