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  3. 대수 및 대수기하 Algebra & Algebraic Geometry

Minimum circumference length of a triangle having each of its three vertices on the sides of an equilateral triangle

작성자 Uploader : 208st 작성일 Upload Date: 2018-10-15변경일 Update Date: 2018-10-17조회수 View : 348

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As shown in the figure, when there are three points P, Q, and R on each side of an equilateral triangle with length L of one side, the minimum circumference length of ΔPQR connecting these three points can be obtained as follows.


If we symmetrically move ΔABC with respect to the side AC and then symmetrically move with respect to the side BC, we get triangles in contact as shown at the bottom of the figure. Here, when the route PP becomes a straight line, the circumference length of ΔPQR becomes minimum.

When the point P is away from the point A by an arbitrary length d, the length of the line segment PP can be obtained as follows.

If the vertical length from A to P is h, then h = (3/4) ^ (1/2) d, which is equal to the vertical length up from the lower right edgeto another point P, and the height of the right triangle with hypotenuse PP It is as follows.

H-2h = H- (3) ^ (1/2) d = (3/4) ^ (1/2) L- (3)

The length of the base of a right triangle is as follows.

2L-e-f = 2L- (L-d) / 2-d / 2 = 2L-

Therefore, the length of line segment PP is as follows.

PP = ((3/4) ^ (1/2) L- (3) ^ (1/2) d) ^ 2 + ((3/2) L) ^ 2)
= ((3/4) L ^ 2 - 3dL + 3d ^ 2 + (9/4) L ^ 2) ^ (1/2)
= (1/2) (12L ^ 2 -12dL + 12d ^ 2) ^ (1/2)
= (3) ^ (1/2) (L ^ 2 - dL + d ^ 2) ^ (1/2)

Assuming L as a constant, L ^ 2 - dL + d ^ 2 has the minimum value at d = L / 2.

By substituting this, the minimum circumference length of ΔPQR can be obtained as follows.

PP = (3) ^ (1/2) (L ^ 2 - L ^ 2/2 + L ^ 2/4)
= (3/2) L

This is the equilateral triangle after the bisector of each side of the equilateral triangle ΔABC.
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■ Minimum circumference length of ΔPQR


■ With the distance d known, minimum circumference length of ΔPQR


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