“청렴 강직한 이해찬…10살 후배지만 민주주의 스승”

Sure, if you provide specific coordinates for the vertices of a triangle, I can help you find the orthocenter. The orthocenter is the point where the altitudes of the triangle intersect, and using the method you've outlined, we can calculate its coordinates step-by-step for your specific triangle. Please provide the coordinates of the vertices \( A(x_1, y_1) \), \( B(x_2, y_2) \), and \( C(x_3, y_3) \), and I'll assist you with the calculations. Finding the coordinates of the orthocenter of a triangle involves determining the intersection point of the altitudes of the triangle. The altitudes are the perpendicular lines drawn from each vertex to the opposite side. Given the vertices of a triangle \( A(x_1, y_1) \), \( B(x_2, y_2) \), and \( C(x_3, y_3) \), here's a step-by-step method to find the orthocenter: 1. **Find the slopes of the sides:** - Slope of \( BC \): \( m_{BC} = \frac{y_3 - y_2}{x_3 - x_2} \) - Slope of \( AC \): \( m_{AC} = \frac{y_3 - y_1}{x_3 - x_1} \) - Slope of \( AB \): \( m_{AB} = \frac{y_2 - y_1}{x_2 - x_1} \) 2. **Find the slopes of the altitudes:** - The altitude from \( A \) is perpendicular to \( BC \). So, its slope is the negative reciprocal of \( m_{BC} \). \( m_{AA'} = -\frac{1}{m_{BC}} \). - The altitude from \( B \) is perpendicular to \( AC \). So, its slope is the negative reciprocal of \( m_{AC} \). \( m_{BB'} = -\frac{1}{m_{AC}} \). - The altitude from \( C \) is perpendicular to \( AB \). So, its slope is the negative reciprocal of \( m_{AB} \). \( m_{CC'} = -\frac{1}{m_{AB}} \). 3. **Write the equations of the altitudes:** - Equation of the altitude from \( A \): \( y - y_1 = m_{AA'}(x - x_1) \) - Equation of the altitude from \( B \): \( y - y_2 = m_{BB'}(x - x_2) \) - Equation of the altitude from \( C \): \( y - y_3 = m_{CC'}(x - x_3) \) 4. **Find the intersection of any two altitudes:** - Solve the system of equations given by any two altitudes to find the point of intersection. For example, solving the equations of the altitudes from \( A \) and \( B \) will give you the orthocenter. Let's assume that you've calculated the slopes correctly and substituted back to derive your equations correctly. You'll solve these equations simultaneously (either by substitution, elimination, or using a matrix method if preferred) to find the point where they intersect, which is the orthocenter of the triangle. If you want to provide specific points, I can help you in detail with the calculations for those particular coordinates.