Triangle Geometry
Geometry
Level
3
In a triangle ABC with BC = CA+ (AB)/2. Point P is given on side AB such that BP : P A = 1 : 3.Given that / CAP = m * / CP A, where m is an integer, determine m.
The answer is 2.
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1 solution
Let PC=d and /_APC=Φ. Applying Stewart's Theorem to Tr. ABC with CP as the cevian, we can say that: (3c/4)(b+c/2)²+(c/4)b²=c(3c²/16+d²) which simplifies to d² = b²+ 3bc/4. Now apply the cosine rule in Tr. APC: d²=b²+(3c/4)²-2b(3c/4)cos(A) or b²+ 3bc/4=b²+(3c/4)²-2b(3c/4)cos(A) which yields: cos(A)= 3c/8b - 1/2. Now applying the cosine rule once again in Tr. APC, we can say that: cos(Φ.)= [b²+3bc/4+(3c/4)² - b²]/[2√(b²+3bc/4) (3c/4)]=(b+3c/4)/[2√(b²+3bc/4)]. Hence, cos(2Φ)=2cos²(Φ)-1= 2(b+3c/4)²/4(b²+3bc/4) - 1 which simplifies to 3c/8b - 1/2. In other words, cos(2Φ)=cos(A) or A = 2Φ. which in turn means m=2