Question

X is a point on the side BC of triangle ABC .XM and XN are drawn parallel to AB and AC respectively meeting AB in N and AC in M. MN produced meets CB produced at T. Prove that TX×TX=TB×TC?

Answer 1

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Answer 2

TX×TX=TB×TC

Explanation:

Given information: XM║AB and XN║AC.

According to the Triangle Proportionality Theorem, if a line is parallel to one side of a triangle, then the line divides other two sides proportionally.

Since, XM║AB, by using Triangle Proportionality Theorem we get

$\dfrac{TX}{TB}=\dfrac{TM}{TN}$           .... (1)

Since, XN║AC, by using Triangle Proportionality Theorem we get

$\dfrac{TC}{TX}=\dfrac{TM}{TN}$           .... (2)

From (1) and (2) we get

$\dfrac{TX}{TB}=\dfrac{TC}{TX}$

On cross multiplication we get

$TX\times TX=TB\times TC$

Hence proved.

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