Concurrent Lines

3 cm

Step 1: In an equilateral triangle, the centroid divides each median in ratio 2:1. Step 2: The distance from vertex to centroid = 6 cm (given). Step 3: Since AG:GD = 2:1, and AG = 6 cm, we get GD = 6÷2 = 3 cm. Step 4: In an equilateral triangle, the centroid is also the incentre, so the distance to the opposite side equals the inradius. Step 5: Therefore, the distance from centroid to the opposite side = 3 cm.

A right triangle

In a right triangle, the orthocentre lies at the vertex containing the right angle. If the orthocentre lies on one of the sides, that side must meet at the right-angle vertex, so the triangle is right-angled. In acute triangles the orthocentre is inside, and in obtuse triangles it is outside.

BG = 2GE

Step 1: Recall that the centroid divides each median in the ratio 2:1. Step 2: This means the distance from vertex to centroid is twice the distance from centroid to midpoint. Step 3: For median BE: BG:GE = 2:1, so BG = 2×GE. Step 4: Similarly, AG = 2×GD and CG = 2×GF. Step 5: Therefore, BG = 2GE is the correct statement reflecting the 2:1 division property.

4 cm

Step 1: Understand that the incentre is the centre of the incircle. Step 2: The incircle touches all three sides of the triangle. Step 3: The radius of the incircle is the perpendicular distance from the incentre to each side. Step 4: Since the incircle radius = 4 cm, the distance from incentre to each side = 4 cm. Step 5: This is because the incentre is equidistant from all three sides of the triangle.