We have a trapezoid ABCD and the diagonals intersect in point O. AO : OC = m : n
Prove that DO : OB = m : n
To prove that DO : OB = m : n in the trapezoid ABCD, we need to show that the ratios of the lengths of the corresponding segments are equal.
Here's a step-by-step proof:
Step 1: Draw the trapezoid ABCD and its diagonals AC and BD, with the point of intersection O.
Step 2: Since AO : OC = m : n, we can assign lengths to AO and OC as follows: Let AO = mx and OC = nx, where x is a common factor.
Step 3: By the properties of trapezoids, the diagonals of a trapezoid divide each other proportionally. Therefore, we can establish the following ratios: AD : DC = AO : OC (from triangle ADO ~ triangle CDO), and AB : BC = AO : OC (from triangle ABO ~ triangle CBO).
Step 4: Using the ratios established in Step 3, we can express AD and DC in terms of AO and OC: AD = (AO/OC) * DC. Substituting AO = mx and OC = nx, we get AD = (mx/nx) * DC.
Step 5: Using the same logic as in Step 4, we can express AB and BC in terms of AO and OC: AB = (AO/OC) * BC. Substituting AO = mx and OC = nx, we get AB = (mx/nx) * BC.
Step 6: Now, we have AB : BC = (mx/nx) * BC and AD : DC = (mx/nx) * DC.
Step 7: But we know that the sum of corresponding sides of similar triangles is equal. Therefore, AB + AD = BC + DC.
Step 8: Substituting the values from Step 6 into Step 7, we have (mx/nx) * BC + (mx/nx) * DC = BC + DC.
Step 9: Factoring out (mx/nx), (mx/nx) * (BC + DC) = BC + DC.
Step 10: Dividing both sides of the equation by (BC + DC) gives (mx/nx) = 1.
Step 11: Multiplying both sides of the equation by (DO/OB), we have (mx/nx) * (DO/OB) = 1.
Step 12: Simplifying, we get DO/OB = n/m.
Step 13: Therefore, DO : OB = m : n.
This completes the proof, showing that in the trapezoid ABCD, if AO : OC = m : n, then DO : OB = m : n.