Return to Answer

point1 = {57.7538, 44.0056}; point2 = {57.7607, 43.9983}; line1 = Line[{point1, point2}]; line2 = Line[{{54.3101, 40.715}, {58.4058, 44.624}}]; intersectionPoint = First[{x, y} /. FindInstance[{x, y} \[Element] line1 && {x, y} \[Element] line2, {x, y}]]; Graphics[{ {Gray, line1}, {PointSize[Large], Point[{point1, point2}]}, {PointSize[Large], Red, Point[intersectionPoint]} }, Prolog -> {Dashed, line2}, Frame -> True, ImageSize -> Large ] 

point1 = {57.7538, 44.0056}; point2 = {57.7607, 43.9983}; line1 = Line[{point1, point2}]; line2 = Line[{{54.3101, 40.715}, {58.4058, 44.624}}]; intersectionPoint = First[{x, y} /. FindInstance[{x, y} \[Element] line1 && {x, y} \[Element] line2, {x, y}]]; Graphics[{ {Gray, line1}, {PointSize[Large], Point[{point1, point2}]}, {PointSize[Large], Red, Point[intersectionPoint]} }, Prolog -> {line2}, Frame -> True, ImageSize -> Large ] 

point1 = {57.7538, 44.0056}; point2 = {57.7607, 43.9983}; line1 = Line[{point1, point2}]; line2 = Line[{{54.3101, 40.715}, {58.4058, 44.624}}]; intersectionPoint = First[{x, y} /. FindInstance[{x, y} \[Element] line1 && {x, y} \[Element] line2, {x, y}]]; Graphics[{ {Gray, line1}, {PointSize[Large], Point[{point1, point2}]}, {PointSize[Large], Red, Point[intersectionPoint]} }, Prolog -> Map[{Hue[10 #], Dashing@#, line2} &]@Range[0, .5, 0.01], Frame -> True, ImageSize -> Large ] 

point1 = {57.7538, 44.0056}; point2 = {57.7607, 43.9983}; line1 = Line[{point1, point2}]; line2 = Line[{{54.3101, 40.715}, {58.4058, 44.624}}]; intersectionPoint = First[{x, y} /. FindInstance[{x, y} ∈ line1 && {x, y} ∈ line2, {x, y}]]; Graphics[{ {Gray, line1}, {PointSize[Large], Point[{point1, point2}]}, {PointSize[Large], Red, Point[intersectionPoint]} }, Prolog -> {Dashed, line2}, Frame -> True, ImageSize -> Large ] 

point1 = {57.7538, 44.0056}; point2 = {57.7607, 43.9983}; line1 = Line[{point1, point2}]; line2 = Line[{{54.3101, 40.715}, {58.4058, 44.624}}]; intersectionPoint = First[{x, y} /. FindInstance[{x, y} ∈ line1 && {x, y} ∈ line2, {x, y}]]; Graphics[{ {Gray, line1}, {PointSize[Large], Point[{point1, point2}]}, {PointSize[Large], Red, Point[intersectionPoint]} }, Prolog -> {line2}, Frame -> True, ImageSize -> Large ] 

point1 = {57.7538, 44.0056}; point2 = {57.7607, 43.9983}; line1 = Line[{point1, point2}]; line2 = Line[{{54.3101, 40.715}, {58.4058, 44.624}}]; intersectionPoint = First[{x, y} /. FindInstance[{x, y} ∈ line1 && {x, y} ∈ line2, {x, y}]]; Graphics[{ {Gray, line1}, {PointSize[Large], Point[{point1, point2}]}, {PointSize[Large], Red, Point[intersectionPoint]} }, Prolog -> Map[{Hue[10 #], Dashing@#, line2} &]@Range[0, .5, 0.01], Frame -> True, ImageSize -> Large ] 

More an observation than an answer (I post it here because I cannot put figures in comments). The behaviour seems to be caused by the Dashed attribute.

More an observation than an answer (I post it here because I cannot put figures in comments). The behaviour seems to be caused by the Dashed attribute, and it is the line, not the point, that is not drawn correctly.

This looks like a bug.

Even better, the lower the Dashing, the further away the line is drawn from the intersection point:

point1 = {57.7538, 44.0056}; point2 = {57.7607, 43.9983}; line1 = Line[{point1, point2}]; line2 = Line[{{54.3101, 40.715}, {58.4058, 44.624}}]; intersectionPoint = First[{x, y} /. FindInstance[{x, y} \[Element] line1 && {x, y} \[Element] line2, {x, y}]]; Graphics[{ {Gray, line1}, {PointSize[Large], Point[{point1, point2}]}, {PointSize[Large], Red, Point[intersectionPoint]} }, Prolog -> Map[{Hue[10 #], Dashing@#, line2} &]@Range[0, .5, 0.01], Frame -> True, ImageSize -> Large ] 

I have no idea why this happens.