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$In a Δ ABC, BC = 2, CA = \sqrt{3} + 1 and ∠ C = 60 ° . Feet of the perpendicular$ from A, B and C on the opposite sides BC, CA and AB are D, E and F respectively and are concurrent at P. then which of the following is false?

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$Radius of the circle circumscribing the Δ DEF is \frac{1}{\sqrt{2}}$

$Radius of the circle inscribed in the Δ DEF is \frac{\sqrt{6} - \sqrt{2}}{4}$

$Area of the Δ DEF is \frac{\sqrt{3}}{4}$

$\sin D + \sin E + \sin F = \frac{2 + \sqrt{3}}{2}$

answer is D.

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Detailed Solution

$\begin{aligned} a = 2; b = \sqrt{3} + 1 ∠ C = \frac{π}{3} \\ DEF is pedal triangle of A B C \end{aligned}$

$\begin{array}{l} For triangle DEF, circumradius = \frac{R}{2} \\ In radius = 2 Rcos Acos Bcos C and area = \frac{1}{2} R^{2} \sin 2 Asin 2 Bsin 2 C \end{array}$

$∠ A = 45^{\circ} ∠ B = 75^{\circ} ∠ C = 60^{\circ} R = \sqrt{2} ∠ D = 90^{\circ} ∠ E = 30^{\circ} ∠ F = 60^{\circ}$

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