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April 20, 2020

Triangle and Trigonometry | AIME I, 1999 Question 14

Try this beautiful problem from the American Invitational Mathematics Examination I, AIME I, 1999 based on Triangle and Trigonometry.

Triangle and Trigonometry - AIME 1999


Point P is located inside triangle ABC so that angles PAB,PBC and PCA are all congruent. The sides of the triangle have lengths AB=13, BC=14, CA=15, and the tangent of angle PAB is \(\frac{m}{n}\), where m and n are relatively prime positive integers, find m+n.

Triangle and Trigonometry
  • is 107
  • is 463
  • is 840
  • cannot be determined from the given information

Key Concepts


Triangles

Angles

Trigonometry

Check the Answer


Answer: is 463.

AIME, 1999, Question 14

Geometry Revisited by Coxeter

Try with Hints


First hint

 Let y be the angleOAB=angleOBC=angleOCA then from three triangles within triangleABC we have \(b^{2}=a^{2}+169-26acosy\) \(c^{2}=b^{2}+196-28bcosy\) \(a^{2}=c^{2}+225-30ccosy\) adding these gives cosy(13a+14b+15c)=295

Second Hint

[ABC]=[AOB]+[BOC]+[COA]=\(\frac{siny(13a+14b+15c)}{2}\)=84 then (13a+14b+15c)siny=168

Final Step

tany=\(\frac{168}{295}\) then 168+295=463.

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