2 circles with centers O and O' are drawn to intersect each other at points A and B. Center O of one circle lies on the circumference of the other circle. CD is drawn tangent to the circle with center O' at A. Prove that OA bisects angle BAC.
To prove that OA bisects angle BAC, we can establish that triangle OAC is congruent to triangle O'AC. Once we establish congruence between these two triangles, we can conclude that angle BAO is equal to angle BAO'.
Here's a step-by-step proof:
Step 1: Draw a diagram representing the given information. Label the centers of the circles as O and O', the points of intersection of the circles as A and B, and the point where the tangent CD intersects the circle O' as C.
Step 2: Since CD is tangent to circle O' at point A, angle O'AC is right angle.
Step 3: Let's consider triangle OAC. We know that angle OAC is equal to 90 degrees (since OA is tangent to O'AC).
Step 4: Similarly, in triangle O'AC, angle O'AC is also equal to 90 degrees (as it is the right angle formed by the tangent CD).
Step 5: Since both triangle OAC and O'AC have a right angle and a shared side OA, we can conclude that these two triangles are congruent by using the hypotenuse-leg (HL) congruence criterion.
Step 6: By congruence, we have angle AOC ≡ angle AO'C.
Step 7: Since angles AOC and AO'C are congruent, we can say that angle BAO is equal to angle BAO' (as they are vertically opposite angles).
Step 8: Therefore, we can conclude that OA bisects angle BAC, as angle BAO is equal to angle BAO'.
And that's the proof!
To prove that OA bisects angle BAC, we need to show that the two angles formed by the intersection of OA with the two tangents are equal.
Let's break down the proof step by step:
Step 1: Draw the line segments.
Draw two circles with centers O and O', and let them intersect at points A and B. Draw the tangent CD to the circle with center O' at point A.
Step 2: Identify the relevant angles.
The angle we want to prove is bisected is angle BAC. To do so, we need to look at the angles formed by the tangent CD with OA and OB.
Step 3: Observe the geometry.
Since O is on the circumference of the circle with center O', angle OAO' is a right angle (90 degrees). Therefore, angle O'AB is also 90 degrees, as it is a tangent to the circle.
Step 4: Apply the tangent-chord theorem.
According to the tangent-chord theorem, the angle formed between a tangent and a chord is equal to the angle subtended by that chord in the opposite segment.
In this case, angle CAD is an angle formed between the tangent CD and the chord AB, so it is equal to angle BAC, the angle subtended by AB in the opposite segment.
Step 5: Use the tangent-chord theorem again.
Similarly, angle BAD is an angle formed between the tangent BD and the chord AB. According to the tangent-chord theorem, angle BAD is also equal to angle BAC.
Step 6: Prove the bisector property.
Since we have established that both angles CAD and BAD are equal to angle BAC, we can conclude that OA bisects angle BAC.
By following these steps and applying the tangent-chord theorem, we have proven that OA bisects angle BAC.