Question
The common tangents AB and CD to two circles with centres O and O’ intersect at E between their centres. Prove that the points O, E and O’ are collinear.
✓
Answer
Joint AO, OC and O’D, O’B
Now, in $\triangle\text{EO}'\text{D}$ and,
O’D = O’B [radius]
O’E = O’E [common side]
ED = EB [since, tangentdrawn from an external point to the circle are equal in the length.]
$\therefore\triangle\text{EO'D}\cong\triangle\text{EO'B}$ [by SSS congruence rule]
$\angle\text{EO'D}=\angle\text{EO'B}$
O'E is the angle bisects of $\angle\text{DEB}\ ...\text{(i)}$
Similarily, OE is the angle bisects of $\angle\text{AEC}$
Now, in quadilateral DEBO'
$\angle\text{OD'E}=\angle\text{O'BE}=90^{\circ}$[since, CED is the tangent to the circle and O'D is the radius, i.e,$\text{O'D}\bot\text{CED}$]
$\Rightarrow\angle\text{O'DE}+\angle\text{O'BE}=180^{\circ}$
$\therefore\angle\text{DEB}+\angle\text{DO'B}=180^{\circ}\ ...\text{(ii)}$
[since, DEBo' is cyclic quadilateral]
since, AB is a straight line.
$\therefore\angle\text{AED}+\text{DEB}=180^{\circ}$
$\Rightarrow\angle\text{AED}+180^{\circ}-\angle\text{DO'B}=180^{\circ}$
Now, in $\triangle\text{EO}'\text{D}$ and,
O’D = O’B [radius]
O’E = O’E [common side]
ED = EB [since, tangentdrawn from an external point to the circle are equal in the length.]
$\therefore\triangle\text{EO'D}\cong\triangle\text{EO'B}$ [by SSS congruence rule]
$\angle\text{EO'D}=\angle\text{EO'B}$
O'E is the angle bisects of $\angle\text{DEB}\ ...\text{(i)}$
Similarily, OE is the angle bisects of $\angle\text{AEC}$
Now, in quadilateral DEBO'
$\angle\text{OD'E}=\angle\text{O'BE}=90^{\circ}$[since, CED is the tangent to the circle and O'D is the radius, i.e,$\text{O'D}\bot\text{CED}$]
$\Rightarrow\angle\text{O'DE}+\angle\text{O'BE}=180^{\circ}$
$\therefore\angle\text{DEB}+\angle\text{DO'B}=180^{\circ}\ ...\text{(ii)}$
[since, DEBo' is cyclic quadilateral]
since, AB is a straight line.
$\therefore\angle\text{AED}+\text{DEB}=180^{\circ}$
$\Rightarrow\angle\text{AED}+180^{\circ}-\angle\text{DO'B}=180^{\circ}$
Need a full question paper?
Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.
Explore more
Similar questions
The median of $19$ observations is $30$. Two more observations are made and the values of these are $8$ and $32$. Find the median of $21$ observations taken together.
→Construct a circle with centre 'O' and radius 4.3 cm. Draw a chord AB of length 5.6 cm. Construct the tangents to the circle at point A and B without using centre.
The largest sphere is to be curved out of a right circular cylinder of radius $7\ cm$ and height $14\ cm$. Find the volume of the sphere.
→A two-digit number is such that the product of its digits is $8$. When $18$ is subtracted from the number, the digits interchange their places.
Find the number.
→Find the number.
Write the value of $k$ for which the system of equations $x + ky = 0, 2x - y = 0$ has unique solution.
→$\alpha^2+\beta^2$
→Prove that, any rectangle is a cyclic quadrilateral.
Solve the following quadratic equation by completing the square method.
$m^2-5 m=-3$
→$m^2-5 m=-3$
Write the value of $a_{30} - a_{10}$_ for the A.P. $4, 9, 14, 19, .....$
→Find the roots of the following quadratic equation (if they exist) by the method of completing the square.
$\sqrt{2}\text{x}^2-3\text{x}-2\sqrt{2}=0$
→$\sqrt{2}\text{x}^2-3\text{x}-2\sqrt{2}=0$