A Cevian is a line segment which joins a vertex of a triangle with a point on the opposite side (or its extension). There are 3 special types of cevians and their properties that we will be looking at today.
1. Median
A line joining the midpoint of a side of a triangle to the opposite vertex is called a median.
Properties
- A median divides a triangle into two parts of equal area.
- The point where the three medians of a triangle meet is called the centroid of the triangle.
- Point G is the centroid of \triangle ABC
- The centroid of a triangle divides each median in the ratio 2:1.
- \frac{AG}{GD} = \frac{BG}{GE} = \frac{CG}{GF} = 2
- The three medians divide the triangle into 6 triangles with equal area.
- [AFG] = [BFG] = [BDG] = [CDG] = [CEG] = [AEG]
An Important Result
- 2(AD)^2 + 2 \cdot \big( \frac{BC}{2} \big) = (AB)^2 + (AC)^2
2. Angle Bisector
An angle bisector of a triangle is a segment or ray that bisects an angle and extends to intersect the opposite sides.
Properties
- The angle bisectors meet at a point known as the incenter.
- Point G is the incenter of \triangle ABC.
- The incenter is the center of the incircle of a triangle.
- The 3 sides are all equidistant from the incenter, Point G.
Important Results
- Angle Bisector Theorem: \frac{AB}{BD} = \frac{AC}{CD}.
- Angle Bisector Length Formula: (AD)^2 = AB \cdot AC - BD \cdot DC
3. Altitude
A perpendicular drawn from any vertex to the opposite side is known as the altitude.
Properties
- The altitudes of a triangle meet at a point known as the orthocenter.
- Point G is the orthocenter of \triangle ABC.
- The angle made by any side at the orthocenter and the vertical angle make a supplementary pair.
- \angle A + \angle BGC = 180 ^{\circ}.
Special Cases
4. Perpendicular Bisector
Perpendicular bisector to any side is the line that is perpendicular to that side and passes through its mid-point. Perpendicular bisectors do not need to pass through the opposite vertex.
Properties
- The point at which the perpendicular bisectors of the sides meet is called the circumcenter of the triangle.
- The circumcenter is the center of the circle that circumscribes the triangle. There can only be one such circle for each triangle.
- Angle formed by any side at the circumcenter is twice the vertical angle opposite to the side. This is the property of the circle whereby angles formed by an arc at the center are twice that of the angle formed by the same arc in the opposite arc.
- \angle CAD = 2 \cdot \angle CBD
- The circumcenter is equidistant from the three vertices of the triangle.
- Circumcenter of a right triangle is the midpoint of the hypotenuse.
\boxed{\text{Note: The circumcenter does not have to be inside the triangle}}
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