ABCD is cyclic quadrilateral and AB =CD then prove that AC=BD
Good
To prove that AC = BD in a cyclic quadrilateral ABCD, where AB = CD, we can make use of the properties of cyclic quadrilaterals and some basic geometry.
Here's how you can prove it step by step:
Step 1: Draw the diagram. Label the given points as A, B, C, and D. Make sure the quadrilateral ABCD is cyclic, which means that all four points lie on the circumference of a circle.
Step 2: Since AB = CD (given), we can label their lengths as 'x'. Mark the center of the circle as O.
Step 3: Join AC and BD, creating two line segments. Label the point of intersection as M.
Step 4: By the properties of cyclic quadrilaterals, the opposite angles of the quadrilateral ABCD are supplementary. Therefore, angle BAD = angle BCD, and angle BCD = angle BAC.
Step 5: Similarly, angle CDA = angle CAB.
Step 6: In the triangle ABC, angles BAC and BCA are equal because they are opposite angles of the cyclic quadrilateral.
Step 7: Using the Angle Sum Property of a triangle, we know that angle BAC + angle BCA + angle ABC = 180 degrees. Since angle BAC = angle BCA, we can rewrite it as 2 * angle BAC + angle ABC = 180 degrees.
Step 8: Since angle BAC = angle BCA, we can rewrite the equation as 2 * angle BAC + angle BAC = 180 degrees, which simplifies to 3 * angle BAC = 180 degrees.
Step 9: Solving for angle BAC, we find that angle BAC = 60 degrees.
Step 10: Since angle BAD = angle BAC = 60 degrees and angle BCD = angle BAC = 60 degrees, we have two congruent triangles, ABD and DBC, by the Angle-Side-Angle (ASA) criterion.
Step 11: Congruent triangles have equal corresponding sides. Therefore, AB = BD and CD = BD.
Step 12: Since AB = BD and CD = BD, we can conclude that AB = CD = BD.
Step 13: In triangles ABC and CDA, we have AB = CD = BD and angle BAC = angle CDA.
Step 14: Using the Side-Angle-Side (SAS) criterion, we can conclude that triangles ABC and CDA are congruent.
Step 15: Congruent triangles have equal corresponding sides. Therefore, AC = AD.
Step 16: From step 12 and step 15, we have AB = CD = BD and AC = AD.
Step 17: Since AC = AD and BD = CD, we can conclude that AC = BD.
Thus, we have proved that in a cyclic quadrilateral ABCD where AB = CD, AC = BD.