/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 32 Let \(O_{9}, I, G\) be the 9 -po... [FREE SOLUTION] | 91影视

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Chapter 5: Problem 32

Let \(O_{9}, I, G\) be the 9 -point center, the incenter and the centroid, respectively, of a triangle \(A B C\). Prove that lines \(O_{9} G\) and \(A I\) are perpendicular if and only if \(\widehat{A}=\frac{\pi}{3}\)

Short Answer

Expert verified
In a triangle ABC, show that lines O_9G and AI are perpendicular if and only if 鈭燗 = 蟺/3, where O_9 indicates the 9-point center of the triangle, I indicates the incenter, and G indicates the centroid. Solution: It has been demonstrated that lines O_9G and AI are perpendicular if and only if the condition 鈭燗 = 蟺/3 is met.

Step by step solution

01

Define notation and properties of special points

Let \(O_9\), \(I\), and \(G\) denote the \(9\)-point center, the incenter, and the centroid of \(\triangle ABC\), respectively. Let also \(M_a\) represent the midpoint of segment \(BC\). Recall that the \(9\)-point circle passes through the midpoints of the sides, the feet of the altitudes, and the Euler points of a triangle.
02

Relating AI and M_a with A

Draw line \(AI\), and let it intersect the \(9\)-point circle at points \(E\) and \(F\) (\(I\) lies inside the triangle). \(E\) and \(F\) are the midpoints of some heights of triangle \(ABC\), thus \(M_a\) lies on segment \(EF\). Note that \(\angle AI M_a = \frac{1}{2}\angle A\).
03

Relating G, O_9, M_a, and the 9-point circle

Since \(G\) is the centroid, we have \(G M_a = 2 \cdot GM_b = 2 \cdot GM_c\), where \(M_b\) and \(M_c\) represent the midpoints of segments \(AC\) and \(AB\), respectively. It implies that \(G\) is the centroid of the triangle \(M_a M_b M_c\). Therefore, \(GO_9\) is perpendicular to \(M_a M_b\), and it bisects segment \(M_a M_b\). So we have \(\angle O_9G M_a = \frac{1}{2}\angle M_a=M_a M_b M_c\)
04

Conditions for O_9G and AI being perpendicular

Lines \(O_9 G\) and \(AI\) are perpendicular if and only if \(\angle AI M_a + \angle O_9 G M_a = \frac{\pi}{2}\) or \(\frac{1}{2}\angle A + \frac{1}{2}\angle M_a M_b M_c = \frac{\pi}{2}\). Since \(\angle M_a M_b M_c=\frac{1}{2}\angle A\), the condition is equivalent to \(\angle A = \frac{\pi}{3}\). Thus, lines \(O_9 G\) and \(AI\) are perpendicular if and only if \(\angle A = \frac{\pi}{3}\).

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