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# Using theorem 6.1, prove that a line drawn through the midpoint of one side of the triangle parallel to the other side bisects the third side. (Recall that you have proved it in class IX).Theorem 6.1: If a line is drawn parallel to one side of a triangle to intersect the other two sides in distinct points, the other two sides are divided in the same ratio.  Hint: Think about what we have given and what we can conclude from it. Write down what exactly we need to prove in this problem and use basic proportionality theorems.

In $\vartriangle ABC$, we have given that, D is midpoint of AB and DE is parallel to BC. $\therefore AD = DB$.

We need to prove that AE=EC. Since, $DE\parallel BC$. By basic proportionality theorem, $\dfrac{{AD}}{{DB}} = \dfrac{{AE}}{{EC}}$.
Also, AD=DB so $\dfrac{{AE}}{{EC}} = 1 \Rightarrow AE = EC$. Hence proved. Note: In theorems unlike the other mathematical problems, we need to think logically. What we have given and how we can use it in order to reach our goal.
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Table of 61 - Multiplication Table of 61  Rolle’s Theorem and Lagrange’s Mean Value Theorem  Lagrange Theorem  Basic Proportionality Theorem(BPT)  Factor Theorem  Bayes Theorem  Apollonius Theorem  Stewart’s Theorem  Green’s Theorem  Circle Theorems  