Area of the triangle formed by joining three centers of a circle
Problem Statement: Circles with centers P,Q and R having radii 1,2 and 3 respectively, lie on the same side of line l and are tangent to l at P', Q' and R' respectively, with Q' between P' and R' . The circle with center Q is externally tangent to each of the other two circles. What is the area of triangle PQR?
Solution:
We get the following figure after analyzing the above statements
We need to find the area of the triangle colored in blue. There can be many ways to solve the problem, I am going to solve it using coordinate geometry.
Taking the smallest center of the smallest circle as the origin(0,0) we will find the other two coordinates and then we will apply the area of triangle formula in coordinate geometry.
For the same we need to draw some extra lines and label the points.
Let the center of the smallest circle be O and the middle circle be A and drawing a line parallel to the tangent from O meeting the perpendicular radius at B, now in triangle AOB the length of OA is the sum of two radii 1+2=3 of the circle and the lenght AB is equal to difference of two radii 2-1 = 1 and by applying the pythagoras theorem we get OB as square root of 8.
Now we can see that from O coordinate to A coordinate the x value is increased by square root of 8 unit and y value has been increased by one unit, so the coordinate of A will be
(2*root(2) , 1).
Now, similarly for the coordinate of the center of the third circle,
drawing a line parallel to the tangent from A meeting the perpendicular radius at E, now in triangle DAE the length of DA is the sum of two radii of the circle 2+3=5 and the lenght DE is equal to difference of two radii 3-2 = 1 and by applying the pythagoras theorem we get AE as square root of 24. Now we can see that from A coordinate to E coordinate the x value is increased by square root of 24 unit and y value has been increased by one unit, so the coordinates of D will be (2*(root(2)+ root(6) , 2).
Now we have the three points of the triangle AOD as O (0,0) , A (2*root(2) , 1) and D (2*(root(2)+ root(6)) , 2).
Now applying the area of triangle formula using matrix :
Substituting the values we get:
This problem was asked in AMC 2016, hope you guys liked the article, have a nice day!