在几何中，等腰三角形是具有相等边长的两个边的三角形。有时将其指定为恰好具有相等长度的两个边，有时将其指定为至少具有相等长度的两个边，因此，后者的形式在特殊情况下包括等边三角形。等腰三角形的例子包括等腰直角三角形，金色三角形，双锥面和某些加泰罗尼亚实体。
Theorem Statement: Angle opposite to equal sides of an isosceles triangle are equal.
证明：
Given, an Isosceles triangle ABC, where the length of side AB equals the length of side AC.
Therefore, AB = AC
Construction:
Let us draw the bisector of ∠A
Let D be the point of intersection of this bisector of ∠A and BC.
Therefore ,by construction ∠BAD = ∠CAD.
In ∆BAD and ∆DAC,
AB = AC (Given)
∠BAD = ∠CAD (By construction)
AD = AD (Common side in both triangle)
So, ∆BAD ≅ ∆CAD (By SAS rule)
So, ∠ABD = ∠ACD, since they are corresponding angles of congruent triangles.
So, ∠B = ∠C
Hence, Proved that an angle opposite to equal sides of an isosceles triangle is equal.
Note:
The converse of this theorem is also true. The sides opposite to equal angles of a triangle are also equal.
基于定理的样本问题
问题1：E和F分别是∆ABC的等边AB和AC的中点(请参见给定图)。证明BF = CE。
解决方案：
Given:
Length of side AB = AC
To show: BF = CE
In ∆ABF and ∆ACE,
AB = AC (Given)
∠A = ∠A (Common)
AF = AE (Halves of equal sides)
So, ∆ABF ≅ ∆ACE (SAS rule)
Since, If two triangles are congruent, their corresponding sides are equal.
Therefore, BF = CE ( by CPCT)
问题2：给定∆ABC的周长为13厘米，∠ABC=∠ACB，且BC边的长度等于3 cm。求出AB和AC边的长度。
解决方案：
Given:
BC = 3cm, Perimeter of ∆ABC = 13cm
∠ABC = ∠ACB
Since ∠ABC = ∠ACB , therefore by applying theorem, the sides opposite to equal angles of a triangle are also equal.
So, length of side AB = AC.
Let the side of AB be x.
Therefore, Perimeter = AB + BC + AC
13 = x + 3 + x ( Since, AB = AC )
13 = 2x + 3
13 – 3 = 2x
10/2 = x
Therefore x = 5
So, the length of side AB and AC is 5 cm.