The Midpoint Of The Points Of Tangency

Geometry Level 4

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

Details and assumptions

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

  • Once you find the exact form of kk, you might use a calculator to proceed.

  • The last digit of the answer that is being accepted is 88. If your answer slightly differs from the intended one because of approximation errors, you should enter the closest integer whose last digit is 88.

  • GeoGebra users will be prosecuted.

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