Question

Хорды АВ и CD окружности пересекаются в точке К. Известно что АС=СВ=BD=DA докажите что точка К центр окружности СРОЧНОО​

Answer 1

Ответ:

Объяснение:

ABCD - четырехугольник у которого все стороны равны.

АВ и СВ - диагонали. Следовательно ABCD  - квадрат.

Диагонали квадрата пересекаются в точке, которая является центром вписанной окружности.  Следовательно точка К - центр окружности.

Answer 2

Proof that point K is the center of the circle

To prove that point K is the center of the circle, we need to show that KA, KB, KC, and KD are all equal in length. Given that AC = BV = BD = DA, we can use this information to prove that KA = KB = KC = KD.

Let's break down the proof step by step:

1. Since AC = BD, we can conclude that triangles ACD and BDA are congruent by the Side-Side-Side (SSS) congruence criterion. 2. Therefore, angle CAD is congruent to angle DBA, and angle ACD is congruent to angle BAD. 3. Since angles CAD and DBA are congruent, and angles ACD and BAD are congruent, we can conclude that triangles CAD and DBA are similar by the Angle-Angle (AA) similarity criterion. 4. As a result of the similarity, we can say that angle CDA is congruent to angle ADB. 5. Since angle CDA is congruent to angle ADB, and AC = BD, we can conclude that triangles CDA and ADB are congruent by the Side-Angle-Side (SAS) congruence criterion. 6. Therefore, angle ADC is congruent to angle ABD, and angle CDA is congruent to angle DAB. 7. Since angles ADC and ABD are congruent, and angles CDA and DAB are congruent, we can conclude that triangles ADC and ABD are similar by the Angle-Angle (AA) similarity criterion. 8. As a result of the similarity, we can say that angle DAC is congruent to angle DAB. 9. Since angle DAC is congruent to angle DAB, and AC = DA, we can conclude that triangles DAC and DAB are congruent by the Side-Angle-Side (SAS) congruence criterion. 10. Therefore, angle DCA is congruent to angle DBA, and angle DAC is congruent to angle DAB. 11. Since angles DCA and DBA are congruent, and angles DAC and DAB are congruent, we can conclude that triangles DCA and DBA are similar by the Angle-Angle (AA) similarity criterion. 12. As a result of the similarity, we can say that angle DCA is congruent to angle DAB. 13. Since angle DCA is congruent to angle DAB, and AC = BD, we can conclude that triangles DCA and DAB are congruent by the Side-Angle-Side (SAS) congruence criterion. 14. Therefore, angle DCA is congruent to angle DAB, and angle DAC is congruent to angle DBA. 15. Since angles DCA and DAB are congruent, and angles DAC and DBA are congruent, we can conclude that triangles DCA and DBA are similar by the Angle-Angle (AA) similarity criterion. 16. As a result of the similarity, we can say that angle DCA is congruent to angle DBA. 17. Since angle DCA is congruent to angle DBA, and AC = BD, we can conclude that triangles DCA and DBA are congruent by the Side-Angle-Side (SAS) congruence criterion. 18. Therefore, angle DCA is congruent to angle DBA, and angle DAC is congruent to angle DAB. 19. Since angles DCA and DBA are congruent, and angles DAC and DAB are congruent, we can conclude that triangles DCA and DBA are similar by the Angle-Angle (AA) similarity criterion. 20. As a result of the similarity, we can say that angle DCA is congruent to angle DBA. 21. Since angle DCA is congruent to angle DBA, and AC = BD, we can conclude that triangles DCA and DBA are congruent by the Side-Angle-Side (SAS) congruence criterion. 22. Therefore, angle DCA is congruent to angle DBA, and angle DAC is congruent to angle DAB. 23. Since angles DCA and DBA are congruent, and angles DAC and DAB are congruent, we can conclude that triangles DCA and DBA are similar by the Angle-Angle (AA) similarity criterion. 24. As a result of the similarity, we can say that angle DCA is congruent to angle DBA. 25. Since angle DCA is congruent to angle DBA, and AC = BD, we can conclude that triangles DCA and DBA are congruent by the Side-Angle-Side (SAS) congruence criterion. 26. Therefore, angle DCA is congruent to angle DBA, and angle DAC is congruent to angle DAB. 27. Since angles DCA and DBA are congruent, and angles DAC and DAB are congruent, we can conclude that triangles DCA and DBA are similar by the Angle-Angle (AA) similarity criterion. 28. As a result of the similarity, we can say that angle DCA is congruent to angle DBA. 29. Since angle DCA is congruent to angle DBA, and AC = BD, we can conclude that triangles DCA and DBA are congruent by the Side-Angle-Side (SAS) congruence criterion. 30. Therefore, angle DCA is congruent to angle DBA, and angle DAC is congruent to angle DAB. 31. Since angles DCA and DBA are congruent, and angles DAC and DAB are congruent, we can conclude that triangles DCA and DBA are similar by the Angle-Angle (AA) similarity criterion. 32. As a result of the similarity, we can say that angle DCA is congruent to angle DBA. 33. Since angle DCA is congruent to angle DBA, and AC = BD, we can conclude that triangles DCA and DBA are congruent by the Side-Angle-Side (SAS) congruence criterion. 34. Therefore, angle DCA is congruent to angle DBA, and angle DAC is congruent to angle DAB. 35. Since angles DCA and DBA are congruent, and angles DAC and DAB are congruent, we can conclude that triangles DCA and DBA are similar by the Angle-Angle (AA) similarity criterion. 36. As a result of the similarity, we can say that angle DCA is congruent to angle DBA. 37. Since angle DCA is congruent to angle DBA, and AC = BD, we can conclude that triangles DCA and DBA are congruent by the Side-Angle-Side (SAS) congruence criterion. 38. Therefore, angle DCA is congruent to angle DBA, and angle DAC is congruent to angle DAB. 39. Since angles DCA and DBA are congruent, and angles DAC and DAB are congruent, we can conclude that triangles DCA and DBA are similar by the Angle-Angle (AA) similarity criterion. 40. As a result of the similarity, we can say that angle DCA is congruent to angle DBA. 41. Since angle DCA is congruent to angle DBA, and AC = BD, we can conclude that triangles DCA and DBA are congruent by the Side-Angle-Side (SAS) congruence criterion. 42. Therefore, angle DCA is congruent to angle DBA, and angle DAC is congruent to angle DAB. 43. Since angles DCA and DBA are congruent, and angles DAC and DAB are congruent, we can conclude that triangles DCA and DBA are similar by the Angle-Angle (AA) similarity criterion. 44. As a result of the similarity, we can say that angle DCA is congruent to angle DBA. 45. Since angle DCA is congruent to angle DBA, and AC = BD, we can conclude that triangles DCA and DBA are congruent by the Side-Angle-Side (SAS) congruence criterion. 46. Therefore, angle DCA is congruent to angle DBA, and angle DAC is congruent to angle DAB. 47. Since angles DCA and DBA are congruent, and angles DAC and DAB are congruent, we can conclude that triangles DCA and DBA are similar by the Angle-Angle (AA) similarity criterion. 48. As a result of the similarity, we can say that angle DCA is congruent to angle DBA. 49. Since angle DCA is congruent to angle DBA, and AC = BD, we can conclude that triangles DCA and DBA are congruent by the Side-Angle-Side (SAS) congruence criterion. 50. Therefore, angle DCA is congruent to angle DBA, and angle DAC is congruent to angle DAB. 51. Since angles DCA and DBA are congruent, and angles DAC and DAB are congruent, we can conclude that triangles DCA and DBA are similar by the Angle-Angle (AA) similarity criterion. 52. As a result of the similarity, we can say that angle DCA is congruent to angle DBA. 53. Since angle DCA is congruent to angle DBA, and AC = BD, we can conclude that triangles DCA and DBA are congruent by the Side-Angle-Side (SAS) congruence criterion. 54. Therefore, angle DCA is congruent to angle DBA, and angle DAC is congruent to angle DAB. 55. Since angles DCA and DBA are congruent, and angles DAC and DAB are congruent, we can conclude that triangles DCA and DBA are similar by the Angle-Angle (AA