Question

Need to Show that for any triangle, the angle bisectors intersect. Then, show that the intersection point of the medians, the intersection point of the altitudes, and the intersection point of the angle bisectors lie on the same line.

Answer 1

In , ray AD, ray BF and ray CE are the angle bisectors. Point I is the Point of Concurrence. It is known as 'Incenter'.

The Euler line - An interesting Fact

It turns out that the orthocenter, centroid, circumcenter and incenter of any triangle are collinear - that is, they always lie on the same straight line called the Euler Line, named after its discoverer.

In the below figure The Intersection point of the medians (Centroid 'G'), Intersection point of the altitudes (Orthocenter 'O') and the Intersection point of the angle bisectors (Incenter 'I') lie on a LINE KNOWN AS 'EULER LINE'.

In The above figure:

1) Angle Bisectors and their intersections are BLACK LINES.

2) Median and their intersections are RED LINES.

3) Altitudes and their intersections are BLUE LINES.

4) EULER LINE is highlighted as YELLOW.

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