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Trangles - Proof

In a triangle \( ABC \) , \( \angle A \) is twice of \( \angle B \) and \( a, b \) and \( c \) are respective sides opposite to angles \( \angle A, \angle B \) and \( \angle C \). then prove that \( a^2 = b(b+c) \)

Note by Muzaffar Ahmed
3 years, 3 months ago

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Let D be on BC such that AD bisects \(\angle A\). Then \(\triangle ABD\) is isoceles. Also, \(\triangle ADC\) is similar to \(\triangle BAC\) as all their corresponding interior angles are equal.

By angle bisector theorem CD=\(\frac {ab}{b+c}\). However, since \(\triangle ACD\) is similar to \(\triangle BCA\), \(\frac {a}{b}=\frac {b}{\frac {ab}{b+c}}=\frac {b+c}{a}\). Cross multiplying gives the result.

Joel Tan - 3 years, 3 months ago

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One of the proofs is this.. Does anyone have a simpler proof?

Let \( \angle A = 2x \) , \( \angle B = x \)

From Sine rule,

\( \frac{Sin A}{a} = \frac{Sin B}{b} \)

\( \implies \frac{Sin A}{Sin B} = \frac{a}{b} \)

\( \implies \frac{Sin 2x}{Sin x} = \frac{a}{b} \)

\( \implies \frac{2 Cos x Sin x}{Sin x} = \frac{a}{b} \)

\( \implies 2 Cos x = \frac{a}{b} \) ...(1)

From Cosine rule,

\( Cos A = \frac{b^2 + c^2 - a^2}{2bc} \) ...(2)

\( Cos B = Cos x = \frac{a^2 + c^2 - b^2}{2ac} \) ...(3)

Now, \( Cos A = Cos 2x = Cos^{2} x - Sin^{2} x \)

\( \implies (Cos 2x) + 1 \)

\( = [ Cos^{2} x - Sin^{2} ] + [ Cos^{2} x + Sin^{2} x ] = 2 Cos^{2} x \)

From (1), \( 2 Cos^{2} x = ( Cos 2x ) + 1 = \frac{b^2 + c^2 - a^2}{2bc} + 1 \)

\( = 2 Cos^{2} x = \frac{b^2 + c^2 - a^2 + 2bc}{2bc} \) ...(4)

Dividing (4) with (3),

\( \frac{2 Cos^{2} x}{Cos x} \)

\( = 2 Cos x = \frac{b^2 + c^2 - a^2 + 2bc}{a^2 + c^2 - b^2} \times \frac{2ac}{2bc} \)

\( = 2 Cos x = \frac{b^2 + c^2 - a^2 + 2bc}{a^2 + c^2 - b^2} \times \frac{a}{b} \)

From (1), \( 2 Cos x = \frac{a}{b} \)

\( \implies \frac{a}{b} = \frac{b^2 + c^2 - a^2 + 2bc}{a^2 + c^2 - b^2} \times \frac{a}{b} \)

\( \implies \frac{b^2 + c^2 - a^2 + 2bc}{a^2 + c^2 - b^2} = 1 \)

\( \implies a^2 + c^2 - b^2 = b^2 + c^2 - a^2 + 2bc \)

\( \implies 2a^2 = 2b^2 + 2bc \)

\( \implies a^2 = b^2 + bc \)

\( = \boxed{a^2 = b(b+c)} \)

Muzaffar Ahmed - 3 years, 3 months ago

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zaffar u are super

Karthik Akondi - 3 years, 3 months ago

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Nice use of trigonometry, although similar triangles would quicken the process

Joel Tan - 3 years, 3 months ago

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