Two circles intersect at points A and B. A common tangent touches the first circle at point C and the second at point D. Let B be inside the triangle ACD. Let the line CB intersect the second circle again at point E. Prove that AD bisects the angle CAE.
- print Print
- list Cite
Expert Answers
calendarEducator since 2015
write1,099 answers
starTop subjects are Math and Science
Hello!
This problem requires one relatively simple fact:
Consider a tangent line to a circle and a secant line through the point of tangency. Then the measure of arc between the secant and the tangent is twice as much as the angle between a secant and a tangent.
You can look at the simple proof with the link supplied.
Now to the problem. Look at the picture.
Denote the angle BCD as `alpha` and the angle BDC as `beta`. By the above statement the angle CAB is also `alpha` (it is based on the arc CB and also a half of it). Also, the angle DAB is `beta` (it is based on the arc BD of the second circle).
Next, tha angle DBE is `alpha+beta` as an external angle of a triangle CBD. And the angle DAE is equal to the angle DBE because they have the same base DE in the second circle.
So both CAD and DAE are `alpha+beta,` QED.
(many thanks to my old math teacher N.N.)
Related Questions
- Let A,B,C be three sets. Show that A intersect (B union C) = (A intersect B) union (A intersect C)
- 1 Educator Answer
- Find the point of intersection of the tangents to the curve y = x^2 at the points (-1/2, 1/4) and...
- 2 Educator Answers
- The center of a circle is at (-3,-2) and its radius is 7. Find the length of the chord which is...
- 1 Educator Answer
- Use linear approximation, i.e. the tangent line, to approximate 125.1^(1/3) as follows: Let...
- 1 Educator Answer
- Find the centre of the circle that passes through points A(-7,4) B(-4,5) and C (O,3)
- 1 Educator Answer