# The theorem on the angle bisector of a regular triangle

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In a regular triangle, the angle bisector drawn to any side is its height, median and perpendicular bisector.

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# The angle bisector of a regular triangle. Property 1

# The theorem on the angle bisector of a regular triangle

**Proof of the theorem on the angle bisector**

**Step 1**

**Step 2**

**Step 3**

**Step 4**

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In a regular triangle, the angle bisector drawn to any side is its height, median and perpendicular bisector.

The theorem on the angle bisector of a regular triangle

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Consider equilateral triangle ABC (АВ=ВС=АС).

Let BF, AD, CE be the angle bisectors.

We will prove that they are the heights, medians and perpendicular bisectors.

Proof of the theorem on the angle bisector. Step 1

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As АВ=АС and AD is the angle bisector to base BC, then according to the property of an isosceles triangle, AD is the median and the height.

As AD is perpendicular to side BC and divides it in half, then AD is the perpendicular bisector.

Proof of the theorem on the angle bisector. Step 2

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As АС=ВС and СЕ is the angle bisector to base AB, then according to the property of an isosceles triangle, CE is the median and the height.

As CE is perpendicular to side AB and divides it in half, then CE is the perpendicular bisector.

Proof of the theorem on the angle bisector. Step 3

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As АВ=ВС and BF is the angle bisector to base AC, then according to the property of an isosceles triangle, BF is the median and the height.

As BF is perpendicular to side AC and divides it in half, then BF is the perpendicular bisector.

The theorem is proved.

Proof of the theorem on the angle bisector. Step 4

Definition of an equilateral triangle

The criterion for a regular triangle

The area of an equilateral triangle. Formulas

The perimeter of an equilateral triangle

The height of a regular triangle. Properties

The median of a regular triangle. Properties

The property of the angles of an equilateral triangle

The exterior angle bisectors of an equilateral triangle

Similarity of regular triangle

The circle inscribed into an equilateral triangle. Properties

The circle circumscribed around a regular triangle. The theorems

Symmetry in an equilateral triangle

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