As per title, I'm having trouble understanding how it's possible for a person at 45°N to see the Sun rising in a northeasterly direction for 6 months of the year. Even during the summer solstice, the Sun is circling the Tropic of Cancer which is well south of 45°N, so how come it is seen rising north of 45°N?
[PS: I do understand that this is the case, I'm just not understanding why]
This is probably easiest to grasp by using some planetarium software (like Stellarium), and turn on the equatorial grid as in the images below (note the map distortion makes the "circles" distorted, so it is better to use an interactive map you can move around and explore yourself). You'll see that it is only the celestial equator which intersects the horizon directly at the East and West points. Any circle of declination above $ 0^\circ $ will go past West and intersect the horizon North of the West point. Similarly, any declination circle below $ 0^\circ $ will intersect the horizon South of the West point. So, the Sun will only set directly West when its declination is $ 0^\circ $ (the equinoxes), will set North of that when it's declination is above $ 0^\circ $, and South of that when its below $ 0^\circ $.
Near Vernal Equinox
Near Summer Solstice (Northern hemisphere)
Near Winter Solstice (Northern hemisphere)
To take things a step further. If you look directly North, you'll see some declination circles don't intersect the horizon at all. As you move further North in latitude, the center of all of the circles gets higher and higher, and fewer declination circles will intersect the horizon. Eventually, you'll get to a point where the Sun would be directly on the Northern horizon on the summer solstice, and for anyplace North of this, the Sun will never set that day. If you were to go all the way to the North pole, the Sun won't set until the Sun goes below $ 0^\circ $ declination.
This should be fairly easy to understand from the following pictures. The sun is directly to the right in each one and the earth's orientation matches what it would be for the northern hemisphere summer solstice, autumnal equinox, winter solstice, and vernal equinox respectively. The grid lines help indicate which way is east/west (the lines of latitude) and which way is north/south (the meridians).
You can see in the northern summer, all the points along the red line where sunrise is occurring their local east direction is pointing towards the bottom of the image, so the sun will appear north of east. The amount north of east differs because the local east vector is also pointing varying amounts into or out of the image depending on latitude. Along the arctic circle (at the very top), the local east vector is pointing straight out of the image, so sunrise is effectively directly to the north.
The equinoxes mark the point where the sun will rise directly in the east as you can see below. All the local east directions are pointed directly at the sun. This happens at both equinoxes. So for half the year (between each equinox), the sun will rise in the northeast (and set in the northwest), reaching its extremes at the solstices.
This can be proven mathematically fairly easily, at least on a sphere and neglecting refraction. The algorithm can also be followed in Google Earth Pro with the "circle ruler" tool.
Let us denote your location $Y$ (without loss of generality we assume it is in the northern hemisphere). Then we define the subsolar point $S$, the point on the surface of the Earth where the Sun is directly overhead. This is obviously unique at any moment in time and its latitude is equal to the current solar declination $\delta$. We only need to show that the direction towards the subsolar point is north of east for any sunrise in summer and south of east for any sunrise in winter.
You can see the Sun rising or setting if and only if the subsolar point is exactly 90° away, or about 10000 km on the surface. This set of points is, by definition, a great circle, and we'll call it $g$.
Unless you are at one of the poles, $g$ intersects the equator at two distinct points, and they must be antipodes (exactly on the opposite sides of the Earth). We'll name them $A$ and $B$, with $A$ being the one to the east and $B$ to the west.
Now draw a plane $ABY$ through these two points and your location. The angle between the equatorial plane and the plane $ABY$ is equal to your geographic latitude, and your location is the northernmost point of its intersection with the surface. Therefore, your line of sight $\vec{YA}$ points exactly east and $\vec{YB}$ points exactly west.
Finally, as we have shown, for any sunrise at any time of the year, the subsolar point must lie on circle $g$, and its latitude must be $\delta$. So if
The whole situation depicted for a random place $Y$ (Warsaw to be honest). Green line $\vec{YA}$ points directly east, the direction of sunrise on equinox. The orange one points to the northernmost sunrise ($\delta = +\varepsilon \approx 23.44$, at summer solstice) and the blue one to the southernmost one ($\delta = -\varepsilon \approx -23.44°$, at winter solstice).
You cam see what's happening by using Google Maps. Zoom out until you see a lot of the world. Pick a point (left click) then right click and select measure distance. Right click somewhere on the equator, then drag that point along the equator until the distance from the original point is 6225 miles, a quarter of the Earth's circumference. That point will be the geographical position of sunrise or sunset as seen from the original point, on the equinoxes. You'll notice that the line isn't straight. Then change to globe view and center the display on the midpoint of the distance measuring straight line. Zoom in on the initial point in either view: The measuring line will be at right angles to the meridian.
The apparent position of the sun at sunrise (or sunset) is perpendicular to the terminator. I'm sure you've seen one of those world maps that shows the day/night areas over time and the seasons. On the morning of an equinox, the terminators are vertical (following lines of longitude) and the sun rises due east for everyone, regardless of latitude.
As the sun passes north of the equator, everyone sees it as being in the northeast by some amount at sunrise and northwest at sunset.
