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SMT 2023 Geometry #9

Source:

August 9, 2023
geometry

Problem Statement

Triangle △ABC\vartriangle ABC is isosceles with AC=ABAC = AB, BC=1BC = 1, and ∠BAC=36o\angle BAC = 36^o. Let ω\omega be a circle with center B and radius rω=PABC4r_{\omega}= \frac{P_{ABC}}{4}, where PABCP_{ABC} denotes the perimeter of △ABC\vartriangle ABC. Let ω\omega intersect line ABAB at PP and line BCBC at QQ. Let IBI_B be the center of the excircle with of △ABC\vartriangle ABC with respect to point BB, and let BIBBI_B intersect PQP Q at SS. We draw a tangent line from SS to ⊙IB\odot I_B that intersects ⊙IB\odot I_B at point TT. Compute the length of ST.
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