Points O and F lie on sides AB and BC of triangle ABC, respectively. Prove that lines FO and AC

Points O and F lie on sides AB and BC of triangle ABC, respectively. Prove that lines FO and AC are parallel if AB = 48cm, CB = 32cm, AO = 18cm, BF = 20cm

Consider triangles OBF and ABC. Let’s find the ratio of the sides of the triangles:

1) ОВ = AB – ОА = 48 – 18 = 30 (cm).

OB / AB = 30/48 = 5/8.

2) BF / BC = 20/32 = 5/8.

The ratio of the two sides is equal.

And since the angle B of these triangles is common, the triangles OBF and ABC are similar (on two sides and the angle between them).

This means that the angle BOF is equal to the angle BAC and therefore FО is parallel to AC (the angles BOF and BAC are equal as respectively).