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Find the circumcenter using the perpendicular bisectors. Type your coordinates below in (x,y) format.

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I'm sorry, but I can't see images. However, I can guide you on how to find the circumcenter of a triangle using perpendicular bisectors.

  1. Identify Points: Let’s say you have three points A, B, and C in a triangle.

  2. Find Midpoints: Calculate the midpoints of at least two sides. For example, if A = (x1, y1) and B = (x2, y2), the midpoint M1 of AB is: \[ M1 = \left( \frac{x1 + x2}{2}, \frac{y1 + y2}{2} \right) \]

  3. Calculate Slopes: Determine the slopes of the sides of the triangle. The slope of side AB is: \[ \text{slope of AB} = \frac{y2 - y1}{x2 - x1} \]

  4. Find Perpendicular Slopes: The slope of the perpendicular bisector of side AB is the negative reciprocal of the slope of AB.

  5. Write the Equation: Use the midpoint and the perpendicular slope to write the equation of the perpendicular bisector. For example, if the perpendicular slope is m, the equation can be written in point-slope form: \[ y - M1_y = m (x - M1_x) \]

  6. Repeat for Another Side: Do the same for another side, say AC or BC.

  7. Find Intersection Point: The circumcenter is the point where these two perpendicular bisectors intersect. You can solve the two equations simultaneously to find the coordinates (x, y).

If you provide the coordinates of the triangle's vertices, I can help you calculate the circumcenter coordinates step-by-step!