Question

In the given figure , D is a point on the side BC of `Delta ABC` such that `angle ADC=angle BAC`. Prove that `CA^(2)=CBxxCD`.

Answer 1

GIVEN `A Delta ABC` in which D is point on BC such that `angle ADC= angle BAC`.
TO PROVE `CA^(2)=CBxxCD`.
PROOF In `Delta ABC and Delta DAC`, we have
`angle BAC= angle ADC` (given)
`angle ACB= angle DCA` ( common)
`:. angle ABC~ Delta DAC` [ by AA-similarity]
So, the sides of `Delta ABC and Delta DAC` are proportional
`:. (CA)/(CB)=(CD)/(CA)`
Hence, `CA^(2)=CBxxCD`.