One of the first compositional rules that we learn is the rule of thirds. It's relatively simple but definitely effective: divide the frame into three, horizontally and vertically, and use the divisions to place your subject. But rules are made to be broken—once you understand them properly, that is—or at least adapted and challenged. If you're looking to leave behind the rule of thirds but still want place your faith in geometrically validated subject-placement, try the golden triangle.
Determining the golden triangle
Draw an imaginary diagonal line across your frame. Now draw imaginary lines from the other two corners, which each meet the long line at right angles. It should look something like this:
Your points-of-interest are where the lines meet. Use them to place your focal point, for example the eyes in portraits, and use the lines to divide your frame and draw the eye to the focal point to help create dynamic images.
Why use the golden triangle?
On a mundane and practical level, it's easier for some people to visualise the triangle than it is the rule of thirds. Moving towards a more creative purpose, by using triangles to compose your frame you're introducing a strong compositional shape to it with a great sense of balance pitted against a precarious point. And triangles have a nifty way of retaining the attention of the eye within the frame: the eye moves from one point to another in a continuous loop.
Quite specifically with the golden triangle, you give yourself a means of dividing the frame in a way that is frequently more pleasing to the eye than a horizontal or vertical split. As well as using the lines to draw the eye to focal points, the use of triangles in the frame brings balance to the image. Think of one half as blue and the other as yellow. Or shadow versus light.
By counter-poising the two points-of-interest against each other, you can enhance the sense of balance in the frame. You get dynamism and balance in one go: brilliant.
Putting it into practice
It's all very well knowing the theory - what about the practice? Try portraits with your subject leaning into the frame and the eyes on a point-of-interest. Use the rule to place bridges in your frame, and have the eye travel along them to a focal point. Just give it a try - you never know!