Question

Given, $A{A_1}$ ,$B{B_1}$ and $C{C_1}$ are the medians of triangle $ABC$ whose centroid is G. If points $A$, ${C_1}$, $G$ and ${B_1}$ are concyclic , then
A. $2{b^2} = {a^2} + {c^2}$
B. $2{c^2} = {a^2} + {b^2}$
C. $2{a^2} = {b^2} + {c^2}$
D. $3{a^2} = {b^2} + {c^2}$

Answer 1

Hint: First of all, we shall learn about the meaning and required definition of the median of a triangle. A median of a triangle is nothing but a line segment divides the opposite side of a triangle into two halves); line segment is a part of a line which contains two endpoints and has a definite length.
Let us deal with this concept of the median of a triangle with an example.
Let us consider a triangle ABC, where A, B and C are its vertices.
In the given figure, DA is the line segment joining the vertex A that divides BC into two halves and so AD is the median of the triangle ABC.

The points which all lie on the same circle are said to be concyclic points.
Now, four or more that lie on a circle are called the concyclic triangles.

Complete step by step answer:
Using the given information, a diagram is drawn as follows.

\[BG.B{B_1} = B{C_1}.BA\]
\[ \Rightarrow \dfrac{2}{3}B{B_1}.B{B_1} = \dfrac{c}{2}.c\]
\[ \Rightarrow \dfrac{2}{3}{(B{B_1})^2} = \dfrac{{{c^2}}}{2}\]
\[\dfrac{2}{3}(\dfrac{{2{a^2} + 2{c^2} - {b^2}}}{4}) = \dfrac{{{c^2}}}{2}\]
\[ \Rightarrow 2{a^2} + 2{c^2} - {b^2} = 3{c^2}\]
\[ \Rightarrow \;2{a^2} = {b^2} + {c^2}\]

Hence, option C is correct.

Note:
The points which all lie on the same circle are said to be concyclic points.
Now, four or more that lie on a circle are called the concyclic triangles.
A median of a triangle is nothing but a line segment which joins a vertex to the midpoint of the opposite side of the vertex (i.e. we can also say that it is a line segment divides the opposite side of a triangle into two halves). Here, the line segment is a part of a line which contains two endpoints and has a definite length.