https://www.khanacademy.org/math/geometry-home/triangle-properties/perpendicular-bisectors/v/circumcenter-of-a-triangle 2023-05-12T21:01:01.897246564Z Circumcenter of a triangle To prove that any point on the perpendicular bisector of a segment is equidistant from the endpoints, draw the segment and bisector, and note that the bisector creates two right triangles. By the definition of a perpendicular bisector, the hypotenuses of these triangles are congruent. Using the Pythagorean theorem, show that the two right triangles are congruent, and thus the point on the bisector is equidistant from the two endpoints. Other proofs are also included in the video. Sal Khan video https://cdn.kastatic.org/googleusercontent/xL2az4HvdZHS0YBW4phKCkMpmvhPYY5BQTn26YnDjWv7fzsHx7w4M33DNflNW4JszKvm0OWEM4OrxBizWV3Sew74 Circumcenter of a triangle To prove that any point on the perpendicular bisector of a segment is equidistant from the endpoints, draw the segment and bisector, and note that the bisector creates two right triangles. By the definition of a perpendicular bisector, the hypotenuses of these triangles are congruent. Using the Pythagorean theorem, show that the two right triangles are congruent, and thus the point on the bisector is equidistant from the two endpoints. Other proofs are also included in the video. https://cdn.kastatic.org/ka-youtube-converted/KXZ6w91DioU.mp4/KXZ6w91DioU.mp4 749 Elementos de un triángulo https://www.khanacademy.org/math/geometry-home/triangle-properties/perpendicular-bisectors/v/circumcenter-of-a-right-triangle 2020-07-30T12:57:49Z Circumcenter of a right triangle Showing that the midpoint of the hypotenuse is the circumcenter Sal Khan video https://cdn.kastatic.org/googleusercontent/-mALeLxggAhnSIJ9-v9bzSqiQdD9v0Z5nbBTtU1wDp6u1MO8ezEfPHnvgJCFg0lx5Cz6-872RBdPUiPCEB4Khzoj Circumcenter of a right triangle Showing that the midpoint of the hypotenuse is the circumcenter https://cdn.kastatic.org/ka-youtube-converted/VejCw2NlE60.mp4/VejCw2NlE60.mp4 342 Elementos de un triángulo https://www.khanacademy.org/math/geometry-home/triangle-properties/angle-bisectors/v/incenter-and-incircles-of-a-triangle 2023-05-12T21:01:01.897246564Z Incenter and incircles of a triangle The incenter of a triangle is the point at which the three angle bisectors intersect. To locate the incenter, one can draw each of the three angle bisectors, and then determine the point at which they all intersect. The incenter is also notable for being the center of the largest possible inscribed circle within the triangle. Sal Khan video https://cdn.kastatic.org/googleusercontent/6V5pwBUyeQneChMmUYugkzgrml6rBLE7AHW_EOhjzm5d2fUoYW74FL2lNPHk7YGCUprCnnLcRbpm955jmnwp-A4q Incenter and incircles of a triangle The incenter of a triangle is the point at which the three angle bisectors intersect. To locate the incenter, one can draw each of the three angle bisectors, and then determine the point at which they all intersect. The incenter is also notable for being the center of the largest possible inscribed circle within the triangle. https://cdn.kastatic.org/ka-youtube-converted/21vbBiCVijE.mp4/21vbBiCVijE.mp4 478 Elementos de un triángulo https://www.khanacademy.org/math/geometry-home/triangle-properties/medians-centroids/v/triangle-medians-and-centroids 2023-05-12T21:01:01.897246564Z Triangle medians & centroids The centroid of a triangle is the point at which the three medians intersect. To locate the centroid, draw each of the three medians (which connect the vertices of the triangle to the midpoints of the opposite sides). It is referred to as the "center of mass" or "balance point" of the triangle. Centroid divides medians in a ratio 2:3. Sal Khan video https://cdn.kastatic.org/googleusercontent/vioeDcVqvPmfqzHusIO_X9rz2x_bSZDVCDH7IZy7JCOYO_Az000hxgKyQlGb5PxTxxe0YjjBESSyNMYNAc0vlqcR Triangle medians & centroids The centroid of a triangle is the point at which the three medians intersect. To locate the centroid, draw each of the three medians (which connect the vertices of the triangle to the midpoints of the opposite sides). It is referred to as the "center of mass" or "balance point" of the triangle. Centroid divides medians in a ratio 2:3. https://cdn.kastatic.org/ka-youtube-converted/GiGLhXFBtRg.mp4/GiGLhXFBtRg.mp4 539 Elementos de un triángulo https://www.khanacademy.org/math/geometry-home/triangle-properties/medians-centroids/v/medians-divide-into-smaller-triangles-of-equal-area 2021-04-19T14:43:38Z Dividing triangles with medians Showing that the three medians of a triangle divide it into six smaller triangles of equal area. Brief discussion of the centroid as well Sal Khan video https://cdn.kastatic.org/googleusercontent/ilL_4FfwmsiDJItqB7yY3SuAZukiqTOO8gZIUAwTwvLN7EXfTNSgmJcoNw5OUtcwvE8PIYahenAGdB93AFgzsniC Dividing triangles with medians Showing that the three medians of a triangle divide it into six smaller triangles of equal area. Brief discussion of the centroid as well https://cdn.kastatic.org/ka-youtube-converted/Xt4sT0fV9Pw.mp4/Xt4sT0fV9Pw.mp4 466 Elementos de un triángulo https://www.khanacademy.org/math/geometry-home/triangle-properties/medians-centroids/v/proving-that-the-centroid-is-2-3rds-along-the-median 2020-06-23T07:22:43Z Centroid & median proof Showing that the centroid divides each median into segments with a 2:1 ratio (or that the centroid is 2/3 along the median) Sal Khan video https://cdn.kastatic.org/googleusercontent/1aJKdHzTZK1vWjhs7MSxLglgzPbtEMTMPQ2PRJcJ35qCG_QZcN4w07hDe1GHe2fWVxwViLCowe00Nvpt7eV7i3g Centroid & median proof Showing that the centroid divides each median into segments with a 2:1 ratio (or that the centroid is 2/3 along the median) https://cdn.kastatic.org/ka-youtube-converted/i0VS3eEGjiQ.mp4/i0VS3eEGjiQ.mp4 416 Elementos de un triángulo https://www.khanacademy.org/math/geometry-home/triangle-properties/medians-centroids/v/median-centroid-right-triangle-example 2020-06-23T07:22:42Z Median, centroid example Example involving properties of medians Sal Khan video https://cdn.kastatic.org/googleusercontent/IqoJ449DImRbVAeH--EHnZ4zirY9TBwOAtuoEvuXKVIO5IhK2ty-Y1mPW1r6G2Z5Y9NPtXtki_36sv4aDMzQ13ohOw Median, centroid example Example involving properties of medians https://cdn.kastatic.org/ka-youtube-converted/k45QTFCHSVs.mp4/k45QTFCHSVs.mp4 540 Elementos de un triángulo https://www.khanacademy.org/math/geometry-home/triangle-properties/altitudes/v/proof-triangle-altitudes-are-concurrent-orthocenter 2021-04-22T15:00:16Z Proof: Triangle altitudes are concurrent (orthocenter) Showing that any triangle can be the medial triangle for some larger triangle. Using this to show that the altitudes of a triangle are concurrent (at the orthocenter). Sal Khan video https://cdn.kastatic.org/googleusercontent/A6sTQFaB8fZPi2RZaO6C6GWGaJz-Fra7PDD4D8SJy8wnOLYI3sjQ0VuqW8WJ6-_s1ReVtT-V1dW2BVvQ2srn9hOJ3g Proof: Triangle altitudes are concurrent (orthocenter) Showing that any triangle can be the medial triangle for some larger triangle. Using this to show that the altitudes of a triangle are concurrent (at the orthocenter). https://cdn.kastatic.org/ka-youtube-converted/aGwT2-RERXY.mp4/aGwT2-RERXY.mp4 601 Elementos de un triángulo https://www.khanacademy.org/math/geometry-home/triangle-properties/altitudes/v/common-orthocenter-and-centroid 2023-05-12T21:01:01.897246564Z Common orthocenter and centroid If the orthocenter and centroid are the same point, then the triangle is equilateral. This is because the orthocenter is the intersection of the altitudes, which are also the medians and the angle bisectors in an equilateral triangle. The centroid is the intersection of the medians, which divide the triangle into six congruent triangles. Therefore, the orthocenter and the centroid coincide only when the triangle has three equal sides and angles. Sal Khan video https://cdn.kastatic.org/googleusercontent/eAiCjI59nXXUpRC-vGgs5BGCdQ2abGCDZ1Qj7-1tA25a6-5-jOCkQiI76_-eph9OGftNuiU-vwv-Od4dPnvfUks Common orthocenter and centroid If the orthocenter and centroid are the same point, then the triangle is equilateral. This is because the orthocenter is the intersection of the altitudes, which are also the medians and the angle bisectors in an equilateral triangle. The centroid is the intersection of the medians, which divide the triangle into six congruent triangles. Therefore, the orthocenter and the centroid coincide only when the triangle has three equal sides and angles. https://cdn.kastatic.org/ka-youtube-converted/pDZIvyeqX1o.mp4/pDZIvyeqX1o.mp4 385 Elementos de un triángulo https://www.khanacademy.org/math/geometry-home/triangle-properties/triangle-property-review/v/review-of-triangle-properties 2020-06-23T07:22:41Z Review of triangle properties Comparing perpendicular bisectors to angle bisectors to medians to altitudes. Sal Khan video https://cdn.kastatic.org/googleusercontent/s6RQeRvzfr7hUKV5GrUjWyjdj_KCL8ee3E7hyfGbuq2Afe6jmekZCrgVoqGpa9f4LeP3lvb6M_TKw2WGTwXBHK8p Review of triangle properties Comparing perpendicular bisectors to angle bisectors to medians to altitudes. https://cdn.kastatic.org/ka-youtube-converted/KUhdMbx5ges.mp4/KUhdMbx5ges.mp4 597 Elementos de un triángulo