1957 AMC 12 第 26 题

Question

从三角形内部一点向三个顶点引线段。这样形成的三个三角形面积相等的充要条件是该点为：

From a point within a triangle, line segments are drawn to the vertices. A necessary and sufficient condition that the three triangles thus formed have equal areas is that the point be:

内切圆的圆心

the center of the inscribed circle

外接圆的圆心

the center of the circumscribed circle

使该点处形成的三个角都为

120^\circ

such that the three angles formed at the point each be

120^\circ

三角形三条高的交点

the intersection of the altitudes of the triangle

三角形三条中线的交点

the intersection of the medians of the triangle

答案：E

知识点：重心三角形面积中线（几何）

难度评级：1340

小提示：

回想中线如何分割三角形的面积

Recall how the medians partition a triangle’s area

大提示：

三条中线的交点将三角形分成六个小三角形，可两两组合成三个等面积区域

The intersection of the medians divides the triangle into six small triangles paired into three equal-area regions

解答：

三条中线交于重心，并把三角形分成六个等面积的小三角形。以重心和原三角形的一条边为顶点的三个三角形，各由其中两个小三角形组成，所以面积相等。反过来，三个面积相等会给出相等的重心坐标，从而唯一确定重心。

因此，所求的充要点是三条中线的交点，正确答案是 E。

The three medians meet at the centroid and divide the triangle into six small triangles of equal area. Each of the three triangles having the centroid and one side of the original triangle consists of two of those small triangles, so their areas are equal. Conversely, equal areas give equal barycentric coordinates, which uniquely locate the centroid.

Thus, the necessary and sufficient point is the intersection of the medians, and the correct answer is E.

1957 AMC 12 第 26 题

答案：E

难度评级：1340

小提示：

大提示：

解答：

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