Prove line segment BP is 2/3 of line segment BD in triangle ABC

To prove that line segment BP is two-thirds of line segment BD in triangle ABC, we can use the concept of similar triangles.

Step 1: Draw the triangle ABC on a piece of paper.

Step 2: Extend line segment BP to intersect with side AC at point E.

Step 3: By using the property of similar triangles, we need to show that triangle BPE is similar to triangle BDA.

Step 4: To prove the similarity, we need to show that the corresponding angles of both triangles are equal, and the corresponding sides are proportional.

Step 5: Since triangle BPE is formed by extending line segment BP to intersect with side AC, we have two corresponding angles that are equal: Angle BPE is equal to Angle BDA (both right angles), and Angle PBE is equal to Angle ABD (since they are vertical angles).

Step 6: Now, let's focus on the sides. We already know that line segment BP is a common side in both triangles. We need to prove that the other sides are proportional.

Step 7: By using the concept of similar triangles, we know that corresponding sides are proportional in similar triangles. Therefore, we need to show that line segment PE is proportional to line segment AD.

Step 8: To prove that line segment PE is two-thirds of line segment AD, we can use proportionality in triangles. We have BE/BA = PE/AD.

Step 9: Since angle B and angle E are vertical angles, they are equal. This implies that triangle BAE is isosceles, meaning that BE = BA.

Step 10: Substituting BE with BA in the proportion, we get BA/BA = PE/AD, which simplifies to 1 = PE/AD.

Step 11: This shows that line segment PE is equal in length to line segment AD, or in other words, PE = AD.

Step 12: Since we want to prove that line segment BP is two-thirds of line segment BD, we need to find the relationship between line segment PE and line segment BD.

Step 13: By the transitive property, we know that if PE = AD and AD = BD, then PE = BD.

Step 14: This proves that line segment BP (which is equal to PE) is indeed two-thirds of line segment BD in triangle ABC.