Trigonometry
Trigonometry
Allen Star Batch
Grade 11
Question:
In an acute angled triangle $ABC$, $\angle A = 20°$, let $DEF$ be the feet of altitudes through $A, B, C$ respectively and $H$ is the orthocentre of $\triangle ABC$. Find $$\frac{AH}{AD} + \frac{BH}{BE} + \frac{CH}{CF}$$
Step-by-Step Solution
Key Concept: Convert the sum of altitude ratios into a product of sines using circumradius relationships.
The sum $\frac{AH}{AD} + \frac{BH}{BE} + \frac{CH}{CF} = \frac{R}{\Delta}(a\cos A + b\cos B + c\cos C) = \frac{R^2}{\Delta}(\sin 2A + \sin 2B + \sin 2C) = \frac{4R^2}{\Delta}\sin A\sin B\sin C = \frac{bc\sin A}{\Delta} = 2$.
Correct Answer: 2