Prove the theorem that states XDR = DYX

Here is the proof for the given theorem. 1.1 Prove the theorem that states XDR = DYX Step 1: Construction. Draw a diameter through the point of tangency D. Let this diameter be DP, where P is a point on the circumference. Join PX. Step 2: Angle between tangent and diameter. The tangent RS is perpendicular to the diameter DP at the point of contact D. RDP = 90^ Step 3: Angle in a semicircle. The angle subtended by a diameter at any point on the circumference is 90^. DXP = 90^ Step 4: Angles in a triangle. Consider DXP. The sum of angles in a triangle is 180^. XDP + DPX + DXP = 180^ Substitute DXP = 90^: XDP + DPX + 90^ = 180^ XDP + DPX = 90^ (1) Step 5: Relate angles at the point of tangency. From Step 2, we know RDP = 90^. We can express RDP as the sum of RDX and XDP. RDX + XDP = 90^ (2) Step 6: Equate expressions for 90^. From equations (1) and (2): XDP + DPX = RDX + XDP Subtract XDP from both sides: DPX = RDX Step 7: Angles in the same segment. Angles subtended by the same arc at the circumference are equal. Both DPX and DYX are subtended by the arc DX. DPX = DYX Step 8: Conclusion. From Step 6 and Step 7, we can conclude: XDR = DYX This proves the Tangent-Chord Theorem. That's 2 down. 3 left today — send the next one.