Try this beautiful problem from the American Invitational Mathematics Examination I, AIME I, 1999 based on Inscribed circle and perimeter.

Inscribed circle and perimeter – AIME I, 1999


The inscribed circle of triangle ABC is tangent to AB at P, and its radius is 21 given that AP=23 and PB=27 find the perimeter of the triangle

Inscribed circle and perimeter
  • is 107
  • is 345
  • is 840
  • cannot be determined from the given information

Key Concepts


Inscribed circle

Perimeter

Triangle

Check the Answer


But try the problem first…

Answer: is 345.

Source
Suggested Reading

AIME I, 1999, Question 12

Geometry Vol I to IV by Hall and Stevens

Try with Hints


First hint

Q tangency pt on AC, R tangency pt on BC AP=AQ=23 BP=BR=27 CQ=CR=x and

Second Hint

\(s \times r =A\) and \(s=\frac{27 \times 2+23 \times 2+x \times 2}{2}=50+x\) and A=\(({(50+x)(x)(23)(27)})\) then from these equations 441(50+x)=621x then x=\(\frac{245}{2}\)

Final Step

perimeter 2s=2(50+\(\frac{245}{2}\))=345.

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