Question

Let s denote the semi-perimeter of a triangle ABC in which BC=a, CA=b, AB=c. If a circle touches the sides BC, CA, AB at D, E, F respectively, prove that BD=s-b.

Answer 1

Hint: Tangents drawn from a point to a circle are always equal.
Perimeter of a triangle is known as the sum of the length of all sides of the triangle. In this question the semi-perimeter of the triangle is given, so first we will find the perimeter of the triangle by using the property of tangents on a circle and then equating it to the given perimeter we will find BD.

Complete step by step answer:
Given
\[
  BC = a \\
  CA = b \\
  AB = c \\
 \]
Now let us assume the length of
\[
  BD = x \\
  DC = z \\
  AE = y \\
 \]
Now since we know the lengths of tangents drawn from an external point to a circle are equal hence we can say
\[
  BD = BF = x \\
  DC = EC = z \\
  AE = AF = y \\
 \]
To better understand this question first draw the diagram for the given question for the given condition

Now since we know that perimeter of a triangle is the sum of length of the external sides of the triangle hence we can write
\[Perimeter\left( {\Delta ABC} \right) = AE + AF + BF + BD + CE + CD\]
Now this can be written as
\[
  Perimeter\left( {\Delta ABC} \right) = y + y + x + x + z + z \\
   = 2\left( {x + y + z} \right) \\
 \]
It is also given in the question that s denote the semi-perimeter of a triangle ABC, so we can say the perimeter of the triangle will be \[ = 2s\]
This can be written as
\[
  Perimeter\left( {\Delta ABC} \right) \\
   \Rightarrow 2\left( {x + y + z} \right) = 2s \\
   \Rightarrow \left( {x + y + z} \right) = s \\
 \]
Now by further solving this equation we can write
\[x = s - \left( {y + z} \right)\]
Now from the diagram we can say \[y + z = AC\]
Therefore we can write the above equation as
\[x = s - AC\]
And since the length of the side AC is given \[AC = b\]and\[x = BD\] hence we can write
\[x = s - b\]
Hence proved

Note: A maximum of two tangent lines can be drawn from an external point to a circle as a tangent line is a straight line and it just touches a the surface of a curve so in case of a circle only two line can be drawn to the both side of the circle and the length of the lines from the point to the surface will always be equal.