Multiple choice

A circle touches the sides $\overline { \mathrm { BC } } , \overline { \mathrm { CA } } , \overline { \mathrm { AB } }$ of $\Delta \mathrm { ABC }$ at points D, E, F respectively. BD = x CE =y , AF = z then area of $\Delta A B C = \sqrt { x y z ( x + y + z ) }$.

  1. True

  2. False
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A Correct answer
Explanation

The area of a triangle with an incircle touching sides at D, E, F is indeed given by the formula sqrt(r_a * r_b * r_c * s) where r_i are the segments from vertices to tangency points. This is a standard geometric property.

AI explanation

Because the circle is an incircle, the tangent lengths from the vertices are equal, so the semi-perimeter s equals x + y + z. The tangent lengths also determine the triangle's sides as x + y, y + z, and z + x, so the area squared by Heron's formula is s(s-a)(s-b)(s-c). Substituting the values gives (x+y+z)(z)(x)(y), meaning the area is the square root of xyz(x + y + z), making the statement true.

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