In an isosceles triangle the bisector of the angle opposite to the base is the altitude and the median.įigure 3 shows an isosceles triangle ABC with sides AC and BC of equal length. Thus, the median CD is the altitude of the triangle ABC. Since these two angles give in sum the straight angle of 180°, each of them is 90°, that is This also means that the corresponding angles ADC and BDC are congruent. This means that the median CD is the bisector of the angle ACB. It implies that the angles ACD and BCD are congruent. Hence, these triangles are congruent, in accordance to the postulate P3 (SSS) of the triangle Thus, the triangles ADC and BDC have two pairs of congruent sides AD = BD, AC = BC and Since CD is a median, the segments AD and BD are congruent. We need to prove that CD is the altitude of the triangle ABC and the angle bisector The segment CD is the median drawn to the base AB of the triangle.
In an isosceles triangle the median drawn to the base is the altitude and the angle bisector.įigure 2 shows an isosceles triangle ABC with sides AC and BC of equal length. In an isosceles triangle the bisector of the angle opposite to the base is the altitude and the median. In an isosceles triangle the median drawn to the base is the altitude and the angle bisector. It is not difficult to prove, in the same manner, two symmetrical statements: Therefore, the corresponding sides are congruent: AD = BD, and CD is the median of the triangle ABC. Hence, these triangles are congruent, in accordance to the postulate P2 (ASA) of the triangleĬongruency (see the lesson Congruence tests for triangles under the current topic in this site). This also means the triangles ADC and BDC have two pairs of congruent angles ACD = BCDĪnd DAC = DBC that include congruent sides AC = BC. It means that the altitude CD is the bisector of the angle ACB. This implies that the angles ACD and BCD are congruent as complementary angles to the angles Therefore, the triangles ADC and BDC are right triangles with congruent angles DAC and DBC. (this was proved in the lesson Isosceles triangles under the current topic in this site).
The angles BAC and ABC are congruent as the angles at the base of the isosceles triangle ABC Since CD is an altitude, the triangles ADC and BDC are right triangles. We need to prove that CD is the median of the triangle ABC and the angle bisector The segment CD is an altitude drawn to the base AB of the triangle. Let ABC be an isosceles triangle with sides AC and BC of equal length ( Figure 1). In an isosceles triangle the altitude drawn to the base is the median and the angle bisector.
An altitude, a median and an angle bisector in the isosceles triangle
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