The Midpoint Of The Points Of Tangency

Geometry Level 4

In right angled \(\triangle ABC\), \(\angle ABC= 90°\) and \(\angle BCA = 60°\). The incircle of \(\triangle ABC\) touches sides \(\overline{BC}\) and \(\overline{AB}\) at points \(D\) and \(E\) respectively. Let \(F\) be the midpoint of \(DE\). \(P, Q, R\) are the feet of perpendiculars from \(F\) on \(BC, CA, AB\) respectively. Let \(k= \dfrac{AB+BC+CA}{PQ+QR+RP} \). Find \(\left \lfloor 100k \right \rfloor \).

Details and assumptions

  • \(\lfloor x \rfloor \) denotes the greatest integer function, i.e. it is the greatest integer not exceeding \(x\). For example, \(\lfloor 2.3 \rfloor = 2, \lfloor \pi \rfloor = 3, \lfloor 5 \rfloor= 5.\)

  • Once you find the exact form of \(k\), you might use a calculator to proceed.

  • The last digit of the answer that is being accepted is \(8\). If your answer slightly differs from the intended one because of approximation errors, you should enter the closest integer whose last digit is \(8\).

  • GeoGebra users will be prosecuted.


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